# Hold ’em and Fold ’em: Towards Human-scale, Feedback-Controlled Soft Origami
Robots111This work is supported by the National Science Foundation under Award
ECCS 1846954

Immanuel Ampomah Mensah, Jessica Healey, Celina Wu
Andrea Lacunza, Nathaniel Hanson, and Kristen L. Dorsey
Institute for Experiential Robotics, Northeastern University, Boston, USA

###### Abstract

An underdeveloped capability in soft robotics is proprioceptive feedback
control, where soft actuators can be sensed and controlled using only sensors
on the robot’s body. Additionally, soft actuators are often unable to support
human-scale loads due to the extremely compliant materials in use. Developing
both feedback control and the ability to actuate under large loads (e.g. 500
N) are key capacities required to move soft robotics into everyday
applications.

In this work, we independently demonstrate these key factors towards
controlling and actuating human-scale loads: proprioceptive (embodied)
feedback control of a soft, pneumatically-actuated origami robot; and
actuation of these origami origami robots under a person’s weight in an open-
loop configuration. In both demonstrations, the actuators are controlled by
internal fluidic pressure. Capacitive sensors patterned onto the robot provide
position estimation and serve as input to a feedback controller. We
demonstrate position control of a single actuator during stepped setpoints and
sinusoidal trajectory following, with root mean square error (RMSE) below 4
mm. We also showcase the actuator’s potential towards human-scale robotics as
an “origami balance board” by joining three actuators into an open-loop
controlled system with a platform that varies its height, roll, and pitch.
_This work contributes to the field of soft robotics by demonstrating closed-
loop feedback position control without visual tracking as an input and
lightweight, soft actuators that can support a person’s weight_. The project
repository, including videos, CAD files, and ROS code, is available at
https://parses-lab.github.io/kresling_control/

##  1 Introduction and Background Knowledge

Figure 1: The origami actuator (a) overview and (b) under fluidic actuator.
Photographs of (c) the low actuator mass, (d) actuators at different scales,
and (e) a rider preparing to step onto the demonstration balance board.

Soft robots may soon be part of daily life, with envisioned applications
ranging from helping humans [1, 2] to holding fragile objects without causing
damage [3]. Reliably controlled, precise, and soft actuators are a critical
part of this vision. Within the wide array of soft actuator designs and
approaches, soft origami actuators fabricated from paper or polymers offer
mechanical compliance and have the advantage of programmable deformation
through a pattern’s folds [4]. Insight gained from early work in origami robot
fabrication, actuation, and sensing approaches [4, 5, 1] spurred interest in
more complex patterns and motions. One common design for origami soft robots
is the Kresling pattern [6, 7, 8], which is a cylindrical origami shape that
is representative of the buckling pattern formed when a cylinder of paper is
twisted and compressed. While the Kresling pattern originated as a paper
structure, it can be 3D printed out of other compliant materials such as
thermoplastic polyurethane (TPU). 3D printing enables Kresling modules to be
fabricated with consistency and ease, and simplifies the process of adjusting
its dimensions. Prior work has applied the Kresling pattern to create crawlers
[9] and multi-segment continuum arms [10] under magnetic, cable-driven, or
fluidic [8] actuation.

In contrast to the wealth of soft mechanics and actuator work present in
origami robotics literature, position sensing and feedback control research
has received less attention. A critical capability will be accurate state
estimation, position control, and force control [11, 12] that will be
essential for integrating soft robots into daily life. While camera-based
tracking [13, 14] is one approach for position or force sensing, it is not
well suited to portable or high-privacy applications. Soft mechanical sensors
(e.g., capacitors [15], resistors [16], or inductors [17]) affixed to the
robot body can provide accurate force and shape estimation without cameras
[18, 19]. With this submission, we present a feedback-controlled one DoF
origami-inspired robot that builds upon our previous findings [20]. The
position of the robot is determined by capacitive sensors adhered to the
actuators (Fig. 1a). We chose capacitance as the sensing modality for its high
sensitivity, potential for integration into soft fabrication processes, and
ability to measure with compact electronics. We use fluidic actuation to drive
the robot contraction (Fig. 1b). Our approach offers greater portability and
mechanical simplicity as the design contains no internal components.

To demonstrate the potential of this actuator in real-world systems, we
assembled three actuators into an open-loop controlled “origami balance
board.” Three actuators with total mass less than 200 g (Fig. 1c) can support
the mass of a 157 cm tall rider (Fig. 1e).

This work contributes to the state-of-the-art in soft robotics proprioceptive
control by:

  1. 1.

designing and optimizing capacitive position sensors using finite element
analysis (FEA);

  2. 2.

programmatically-generating and fabricating Kresling robots of different
scales through 3D printing; and

  3. 3.

demonstrating feedback control using only the capacitive electrodes.

##  2 Design Component Overview

Figure 2: (left) The fluidic control board, further discussed in section 6.2
(right) A photograph of the system components including: Kresling actuators,
the base plate, and pneumatic tubing entering the side of the base frame.

The components of this project include Kresling actuators, capacitive sensors,
and a fluidic control board (Fig. 2). The case study combines three actuators
with two rigid plates to form a balance board with an actuator mass of 182 g
and a total board mass of 8.6 kg for the frame, plates, and actuators. A
pressurized air source is fed into the fluidic control board and separated
into a pneumatic tube for each Kresling actuator. Pneumatic tubing runs from
the fluidic control board to the base of the Kreslings. The first rigid plate
acts as a base plate for the bottom of the Kreslings. The second rigid plate
rests on top of the actuators and serves as the platform for a rider.
Materials required to fabricate the frame design are pneumatic tubing, glues,
sealants, and rigid boards (e.g., plywood sheets) to act as base plates.

##  3 Material Selection

To fabricate our Kresling actuators, we utilized Thermoplastic Polyurethane
(TPU) filament (Ninjaflex Cheetah, Ninjatek)(Fig. 3) in conjunction with a 3D
printer due to its flexible material properties and cost-effectiveness. The
flexibility and conformability of TPU render it an ideal choice for
constructing dynamic origami structures. It is also non-porous and airtight,
which permits actuation with negative and positive fluidic pressure.

One disadvantage of TPU is that it is highly hygroscopic, so print quality
degrades with increased humidity. As a result, we periodically dry the
filament using a commercial filament dryer to ensure optimal print quality.
While alternative materials such as polymers, paper-like substances, or thin
metals could have been considered for manufacturing, they would have entailed
longer lead times, less control over the structure angles, and potentially
higher costs.

Copper and nickel-plated polyester fabric from Adafruit were selected for the
electrodes due to their flexibility and electrical conductivity. The fabric’s
material structure allowed for easy placement between the folds of the
Kresling origami actuator. Conductive fabric also exhibits excellent
mechanical resistance to torsion and tearing, rendering it suitable for
dynamic operations. Several alternatives to copper and nickel-plated polyester
fabric were considered, including conductive polymers, elastomers, and fabrics
with different metal compositions. However, these alternatives either do not
offer high conductivity, exhibit greater weight and reduced flexibility, or
delaminated from the robot during actuation.

Figure 3: A photograph of the flexible TPU filament used for 3D printing.

##  4 Design Optimization

###  4.1 Origami Actuator Structure and Kinematic Model

Previous work has thoroughly investigated the mechanics of the Kresling
pattern [10, 21]. Here, we present a short definition of properties relevant
to the modeling of the Kresling fold. The structure has a single DOF, as
described by linked parameters $L$, which is the distance between the top and
bottom faces, and $\alpha$, which is the rotation between the faces. Four
parameters control the morphology: the distance between top and bottom faces
at full contraction $L_{c}$, the circumscribing radius around the top face
$R$, the angle ratio $\lambda$ between the valley fold and a bottom edge to
the angle between a bottom edge and a line that bisects a bottom face inner
angle, and the number of Kresling polygon edges $N$. The interior angle formed
between the center and one edge of the top face is $\phi=\pi/N$. Fig. 4
represents these parameters on the Kresling structure with views (a) in 2D,
looking up through the bottom face, and (b) in 3D.

These variables are linked to one another [10] through

$L=\sqrt{L_{c}^{2}+2R^{2}\left[\cos(\alpha+2\phi)-\cos(\alpha_{c}+2\phi)\right]},$
(1)

where $L$ and $\alpha$ represent a distance and rotation, respectively,
between the top and bottom faces, and $\alpha_{c}$ = 2$\lambda(\pi/2-\phi)$ is
the rotation at full contraction.

We will refer to three additional dependent parameters to describe the
Kresling structure. At full extension, the structure has length $L_{e}$ and
rotation $\alpha_{e}$. Two triangular sidewall faces, one with an edge on the
top and the other with an edge on the bottom polygon face, intersect to form a
valley fold with dihedral angle $\xi$.

The fold angle $\xi$ is linked to $L$ by finding the angle of intersection
between vectors normal to the sidewall triangular faces $n_{t}$=

$\begin{bmatrix}RL\sin(2\phi)\hat{i}\\\ RL\left(1-\cos(2\phi)\right)\hat{j}\\\
R^{2}\left(\sin(\alpha)(1-\cos(2\phi))+(1-\cos(\alpha))(\sin(2\phi))\right)\hat{k}\\\
\end{bmatrix}$ (2)

and $n_{b}$=

$\begin{bmatrix}RL(\sin(2\phi-\alpha)+\sin(\alpha))\hat{i}\\\
-RL(\cos(2\phi-\alpha)-\cos(\alpha))\hat{j}\\\
R^{2}\left((\cos(2\phi-\alpha)-\cos(\alpha))(\sin(2\phi-\alpha)+\sin(2\phi))\right)\hat{k}-\\\
R^{2}\left((\cos(2\phi-\alpha)-\cos(2\phi))(\sin(2\phi-\alpha)+\sin(\alpha))\right)\hat{k}\\\
\end{bmatrix}$ (3)

where $\hat{i}$, $\hat{j}$, and $\hat{k}$ are unit vectors on the axes $x$,
$y$, and $z$, respectively, such that

$\cos\xi=\frac{\vec{n}_{t}\cdot\vec{n}_{b}}{||n_{t}||||n_{b}||}.$ (4)

We derived the relationship between $L$ and $\xi$ through Eqs. 1$-$4.
Observing that the relationship between fold angle and length is highly linear
via a numerical approach ($R^{2}$ of $99.3\%$ in the range of $L_{c}$ to
$L_{e}$), we used a linear approximation for the inverse kinematics function
$L=0.22\xi+10.4$ mm, where $\xi$ is in degrees.

Figure 4: The parameters of the Kresling structure (a) from the bottom face
looking up and (b) in 3D

###  4.2 Capacitive Sensor Optimization

Capacitive sensing offers high sensitivity and ease of readout with
measurement electronics, making it suitable for displacement sensing in soft
robots. One potential disadvantage of sensing the pose of the origami robot
with capacitors is the relationship between fold angle, electrode distance,
and capacitance. Although the relationship between $\xi$ and $L$ is well
approximated with a linear function, the reciprocal relationship between
electrode distance and capacitance will yield much lower capacitance at large
(more obtuse) fold angles. In a Kresling structure where nearly the entire
sidewall face is covered with an electrode, these super-linear behaviors will
yield a low sensitivity to displacement when the robot is nearly extended.

To obtain a higher sensitivity at larger fold angles, we optimized the
electrode shape. Due to the lack of closed-form solutions for various
electrode shapes, we used FEA (COMSOL v6.0) and a downhill simplex method
implemented in MATLAB [22] (fminsearch) for optimization. This approach does
not rely on an analytical solution of the gradient (e.g. Newton’s method) and
was computationally feasible with FEA (e.g., vs. genetic algorithm). Electrode
shapes were represented by five values that control the horizontal and
vertical placement of points on the electrode face. The optimization function
was:

$\displaystyle\text{maximize }f(x)=\frac{\Delta C_{65}(x)}{C_{10}(x)}$ (5)

where $\Delta C_{65}(x)$ is the capacitance change of a fold moving between
70$\degree$ and 65$\degree$ from a horizontal plane (i.e., $\xi$/2) and is
constructed of the vertices in the subset $x$, which is a subset of all
potential electrode points in $\mathbb{R}^{2}$ bounded by the sidewall face.
$C_{10}(x)$ is the capacitance at 10$\degree$ also constructed of the same
subset vertices $x$.

Figure 5: Finite element electrode modeling. (a) The initial electrode shapes
IC1-IC5 and OC5. The optimized sensitivities are presented in the top left
corners. (b) The simulation results for electrode shapes OC1$-$OC5 and the
triangle electrode. _Inset:_ $\Delta C_{65}$.

We chose five initial conditions (IC) for the electrode shapes (IC1$-$5)
selected from a larger set for variations in initial capacitance and point
placement to optimize our electrode shapes (Fig. 5a). From the optimized
electrode (OC) from each initial condition, we selected the electrode with the
largest change in capacitance at large fold angle for further testing (i.e.,
OC5). Each minimum fractional sensitivity is presented in Fig. 5a, and the
fractional capacitance change ($C-C_{70})/C_{10}$ is presented in Fig. 5b. The
performance of a triangular electrode inset 1.125 mm from the valley fold
($\Delta C_{65}/C_{10}$ of 1.07%) is also shown. Capacitance was also
simulated for OC5 across the full range of $\xi$ from extension to
contraction.

##  5 Fabrication

Figure 6: The fabrication process. (a) 3D printing, (b) the electrodes after
cutting, (c) the assembled robot with tracking markers for optical motion
capture and ground truth

###  5.1 3D Printing

We 3D printed (Creator Pro 2, Flash Forge) the Kresling structure with TPU
filament (NinjaFlex Cheetah 95A) (Fig. 6a). Print settings were a 0.4 mm
nozzle at an extruder temperature of 238 °C, platform temperature of 50 °C,
layer height of 0.18 mm, and print speed of 30 mms-1.

The STLs were sliced in Simplify3D. The detailed settings are listed in Table
1.

Setting | Value
---|---
Nozzle Diameter | 0.4mm
Extrusion Multiplier | 1.09
Use Retraction | Disabled
Layer Height | 0.18mm
Print Speed | 30mm/s
External Thin Wall Type | Allow Single Extrusions
Internal Thin Wall Type | Allow Single Extrusions
Extruder Temperature | 238 °C
Bed Temperature | 50 °C
Table 1: Simplify3D Slicing Settings

Prior to initiating each print, the 3D Printer goes through a preheating
procedure where the 3D printer’s extruder and print bed are heated to their
specified operating temperatures. Fabricating each human-weight Kresling
actuators requires over nine hours for structure and lid, while the smaller
Kresling structure requires three hours.

###  5.2 Sensorization

To form the electrodes, we cut copper and nickel plated polyester fabric
(Adafruit) into the electrode shape with an electronic cutting machine (Maker
3, Cricut) (Fig. 6b). We adhered these electrodes to opposite faces of the
Kresling valley fold with double-sided pressure sensitive adhesive to form a
capacitor. Three capacitors were symmetrically spaced radially around the $z$
axis. The placement of the electrodes was guided by extruded lines on the
sidewalls.

###  5.3 Sensorized Robot Assembly

To actuate and test the robot, we capped the structure with a TPU lid and
cyanoacrylate glue (Loctite 495) and inserted pneumatic tubing through holes
in the cap. Rigid plastic bases were bonded to the top and bottom faces of the
Kresling actuator to enforce rigid bodies for kinematics and optical tracking.
Fig. 6c is a photograph of the robot mounted to a motion capture benchmarking
stand.

###  5.4 Balance Board Assembly

Three Kreslings were adhered radially around the center of a ridig plate using
hot melt adhesive (Fig. 2). Our design utilized plywood boards for base
plates, and an 80/20 frame to provide clearance for the pneumatic tubing to
enter the bottom of the Kresling actuators. Plywood and 80/20 are common and
robust fabrication materials, however the base can be made out of any rigid
material that leaves space for pneumatic tubes to enter the frame. An all-wood
base would reduce the weight and cost of the system. We also used
cyanoacrylate glue (Loctite 495) and epoxy (Xtc-3D High Performance 3D Print
Coating) for joining the lid to the structure sealing the Kresling actuators.
Any glue or sealant compatible with TPU can be used for this purpose.

##  6 Design Process

###  6.1 CAD Models

To fabricate the Kresling robots, we programmatically generated a CAD model
using a custom script that runs on Autodesk Fusion 360’s Python API. The
script generates 3D Kresling models based on a set of adjustable user-input
parameters. In this script, each fold of the actuator is drawn as two sets of
lofted triangles.

To generate the CAD model using the script available through our GitHub
repository:

  1. 1.

The Kresling.py script must be downloaded from the repository. Place the
downloaded .py file into a folder titled “Kresling.”

  2. 2.

Autodesk Fusion 360 must be downloaded from the Autodesk site. Autodesk
permits educational licenses for Fusion 360 to be used for university
research.

  3. 3.

Open Fusion 360 and select “UTILITIES” in the toolbar.

  4. 4.

Open the “ADD-INS” menu and select “Scripts and Add-Ins”.

  5. 5.

Click the green “+” next to “My Scripts” in the “Scripts” tab.

  6. 6.

Navigate to the “Kresling” folder where the .py script lives.

  7. 7.

Hit “Select Folder” to add the script to Fusion 360.

  8. 8.

To run the script, select “Kresling” in the “My Scripts” list and click the
“Run” button.

  9. 9.

When the script runs, a new document opens in Fusion 360. A window with a set
of adjustable parameters allows the user to modify critical dimensions on the
Kresling. The Kresling model automatically updates while the parameters are
altered.

  10. 10.

To generate the Kresling with the selected parameters, hit “OK”.

Figure 7: The parameters of the generated Kresling can be adjusted in Fusion
360

Once the CAD model is generated, these Kresling building blocks can be further
modified in the CAD and altered based on the intended application. When the
object is ready to be 3D printed, it can be exported as an STL:

  1. 1.

Open the “Bodies” dropdown on the feature tree.

  2. 2.

The model is separated into different bodies so each part can be printed
separately (such as the lid and the main Kresling structure). The main
Kresling structure can be identified by its name, which lists the values of
key parameters used to generate the model.

Figure 8: The main Kresling structure is named after the primary parameters
that define its shape

  3. 3.

Right-click one of the bodies to export it as an STL. Select “Save As Mesh.”

  4. 4.

Ensure the “Format” is set to “STL (Binary)”.

  5. 5.

Hit “OK.” Give the STL an appropriate file name and click “Save.”

###  6.2 Fluidic Control Board

Actuator position is driven by a custom fluidic control board (Fig. 2). Fig. 2
is a photo of the custom fluidic control board. A data acquision system (T7,
LabJack, #1) sources PWM signals to the solenoids (VQ110U-6M, Orange Coast
Pneumatics, #6) to drive actuator pressure. The PWM signals at 5 V are
converted to the 12 V solenoid voltage through a MOSFET driver (5648,
Adafruit, #7), and 12 V power is supplied with a switching power converter
(352, Adafruit, not shown). Ground and 12 V signals are distributed via Wago
snap-action connectors (784, Adafruit, #5). An Arduino Uno (#4), I2C
multiplexer (2717, Adafruit, #2), and ported absolute pressure sensors
connected to the actuator working pressure (3965, Adafruit, #3) complete the
pressure measurement circuitry. An I2C multiplexer is required due to hard-
coded I2C addresses on the pressure sensors. Capacitance is read at a rate of
90 Hz with a multi-channel capacitance-to-digital converter (FDC2214 EVM,
Texas Instruments, not shown). The fluidic pressure source (e.g., vacuum) is
supplied off-board by either a compressor or building pressure line.

Both the LabJack and Arduino are connected to the drive computer, along with
the capacitance-to-digital converter, to output PWM signals to the solenoids
and read in pressure and capacitance data. The following section describes the
structure of the ROS nodes for publishing, converting, and storing these data.

###  6.3 ROS Nodes

The PWM values were set by custom ROS [23] nodes that publish capacitance and
calculate length. The major nodes and functions are described below and a ROS
graph is depicted in Fig 9.

Figure 9: A ROS graph of the nodes within the feedback control loop.

  * •

capacitance_node: The capacitance_node oversees the capacitance data
measurement. It receives data from the electrodes and publishes the
capacitance data of all three electrodes on three separate channels on the
capacitance topic.

  * •

length_publisher_node: This node acts as both a subscriber and a publisher. It
subscribes to the capacitance topic, filters the data and then converts the
measurements into an actuator length estimate. The length is then published on
the kresling_length topic.

  * •

labjack_PID_controller: This node implements the control loop logic. It
subscribes to the kresling_length topic and uses the calculated length to set
the output pwm based on the implemented logic for the given application. The
ROS package included in the github repository contains several of these nodes
for open loop and closed loop implementations as well as single and multi-
kresling applications.

  * •

pressure_node: The pressure node reads the measured pressure, converts it to
gauge pressure and publishes the gauge pressure on the pressure topic.
Although pressure measurements are not currently utilized within the control
mechanisms, the function is enabled to support future investigations.

###  6.4 Closed-loop Control

The kinematic model was combined with the FEA model of capacitance ($C_{FEA}$)
with $\xi$ to estimate the relationship between $L$ and capacitance (Fig.
10a). This relationship was represented in the length calculation function as
a third-order polynomial fit.

Figure 10: Open-loop response with contraction. (a) Measured and kinematic
model of top face rotation $\alpha$. (b) FEA and measured capacitance change,
which is zero at the fully extended length.

Because humidity, local electromagnetic fields, and small variations in
electrode placement affect robot capacitance and the match to the FEA model,
we mapped $C_{FEA}$ to the measured capacitance as
$C_{meas}=\kappa^{-1}C_{FEA}$, where $\kappa$ is a factor that scales
$C_{meas}$.

After attempting multiple control approaches, including proportional-integral-
derivative (PID) and discrete-time control with extensive parameter tuning, we
selected discrete-time control due to its high stability and performance in
this system. The value of the PWM signal at time $t$ that was fed to the
solenoid valves on the fluidic control board was

$\text{PWM}_{t}=\text{PWM}_{t-1}+K_{p}\Delta L+K_{d}\frac{\Delta L}{\Delta t}$
(6)

where PWMt-1 is the PWM value of the previous time step, $K_{p}$ is the
proportional gain constant, $K_{d}$ is the derivative gain constant, $\Delta
L$ is the error between the setpoint and the measured contraction, and $\Delta
t$ is the time step. The proportional and derivative gain values were tuned by
hand with values of 0.005 and 0.001, respectively. The center of mass position
and rotation of the actuator were measured using a commercial motion capture
system (V120:Trio, OptiTrack) and served as the ground truth.

##  7 Case Study: Proprioceptive Feedback Control

Proprioceptive feedback control will be an essential capacity in soft robots
to make them portable and used in applications where vision-based tracking is
not viable. The response of the Kresling robot to a time-varying sinusoidal
and stepped control setpoints is shown in Fig. 11. RMSE is below 4 mm for
setpoint contractions up to 18 mm. The robot uses the change in capacitance
due to displacement to transition between setpoints and track a commanded sine
signal. The video for this case study is available at
https://parses.sites.northeastern.edu/files/2023/10/Kresling_RAL.mp4.

Figure 11: Feedback control response. (a) The response to a commanded
sinusoidal input and (b) multiple control input steps.

##  8 Case Study: Origami Balance Board

The assembled two-dimensional “Origami Balance Board” can serve as a
lightweight and portable platform that is capable of supporting the weight of
a person. This can be utilized in exercise and physiotherapy routines,
potentially aiding in balance training, muscle strengthening and refining fine
motor skills. Similar commercialized boards are passive and cannot adjust to
the needs of a user as they build strength or improve balance through use.

Fig. 12 illustrates the ability of the Kresling actuators in this envisioned
application. The commanded position of each Kresling is randomly modified by
decreasing the internal pressure supplied to the actuator below the working
pressure of $\approx$ 35 psi. Under the rider’s weight, the actuator then
contracts and the top platform tilts. Supplying increased pressure again
raises the rider to full, level height. The video for this case study is
available at:
https://parses.sites.northeastern.edu/files/2023/11/Case_balance.mp4

Figure 12: Photographs illustrating the motion of the balance board and rider

##  9 Contribution Significance

Our project demonstrates soft, 3D-printed origami actuators with demonstrated
proprioceptive feedback control or the ability to support human-scale loads.
In one case study, we demonstrated feedback position control without visual
tracking, using only capacitive electrodes and discrete-time control. These
allowed the actuator to reach setpoint states and track sinusoidal
trajectories. In the second case study, we demonstrated a practical
application of an active balance board, which required three Kresling
actuators (total mass of 183.6 g). These actuators were able to support and
manipulate a rider of approximately 60 kg. We anticipate demonstrating the
ability simultaneous support large loads and proprioceptively control actuator
position in future work.

Integrating soft robots into daily life will require two types of translation:
translation of real world needs into soft robot capabilities, which we
demonstrated through the case studies, and translation of soft robot designs
between researchers to permit deeper research investigations into soft
actuator phenomena and capabilities. Through this design document, we detailed
our progress in designing soft, origami-inspired actuators, including publicly
available code to generate the origami structures; a revised kinematic model
that builds on previous work in origami actuators; an optimized design for
proprioceptive capacitive sensors; a fabrication process that requires only an
FDM printer, computer controlled cutter, and benchtop space; designs for a
control board capable of positive or negative fluidic pressure; and a set of
ROS nodes for controlling actuator position. We anticipate that this
documentation will permit other soft robotics researchers to use the basic
origami actuator structure for their own investigations and applications.

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# Detour Monophonic Vertex Cover Pebbling Number $(DMVCPN)$ of Some Standard
Graphs

K. Christy Rani1, I. Dhivviyanandam2

###### Abstract.

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$.
Pebbling shift is a deletion of two pebbles from a vertex and a placement of
one pebble to a neighbouring vertex. The vertex cover set, $D_{vc}$ for graph
$G$ is the subset of $V(G)$ such that every edge in $G$ has minimum one end in
$D_{vc}$. A detour monophonic path is considered to be a longest chordless
path between two non adjacent vertices $x$ and $y$. A detour monophonic vertex
cover pebbling number, ${\mu}_{vc}(G),$ is a minimum number of pebbles require
to cover all the vertices of the vertex cover set of $G$ with at least one
pebble each on them after the transformation of pebbles by using detour
monophonic paths. We determine the detour monophonic vertex cover pebbling
number (DMVCPN) of Cycle, Path, Fan and Wheel graphs.

Keywords: detour monophonic pebbling, vertex cover set, detour monophonic
vertex cover pebbling.

AMS Subject Classification: 05C12, 05C25, 05C38, 05C76, 05C99

1 Department of Mathematics, Bon Secours College for Women (Autonomous),
Thanjavur-613006,Tamil Nadu, India.
e-mail<EMAIL_ADDRESS>https://orcid.org/0000-0002-2837-9388.
2 Department of Mathematics, North Bengal St. Xavier’s College,
Rajganj-735134, West Bengal, India.
e-mail<EMAIL_ADDRESS>https://orcid.org/0000-0002-3805-6638.

##  1\. Introduction

Graph pebbling is an interesting area emerged a few decades ago and enhanced
Graph theory. Graph pebbling has produced results in Combinatorial Group
Theory and Combinatorial Number Theory [4], [5]. It is widely applied in
network optimization model for transportation and for transmitting information
through its medium to military troops, etc. A pebbling move is an extraction
of two pebbles from one vertex and placing one pebble on the adjacent vertex
and eliminating the other pebble. A vertex cover of a graph $G$ is studied in
[9], which defines the vertex cover as a set $D_{vc}\subset V(G)$ such that
for each edge $uv\in E(G)$, at least one of $u$ or $v$ is in $D_{vc}$. A
detour monophonic path from $x$ to $y$ is the longest path with no chords
studied in [10].In this article we define the detour monophonic vertex cover
pebbling number, ${\mu}_{vc}(G)$, which is a minimum number of pebbles require
to cover all the vertices of the vertex cover set $D_{vc}$ of $G$ and the
corresponding edges of $G$ with at least one pebble each on them after the
transformation of pebbles by using detour monophonic paths. We also
investigate the detour monophonic vertex cover pebbling number of standard
graphs. To prove the worst condition, we use the stacking theorem. It states
that when the initial configuration $D$ is placed on a single vertex $v$ such
that the distance of $v={\dot{s}}$ is maximum and hence ${\dot{s}}=\sum_{u\in
V{G}}{2^{dist(u,v)}}$. Applying this for every vertex $v\in V(G)$, we
determine $\mu_{vc}(G)$. We refer to [1] and [2] for basic terminologies of
graph theory. Throughout this paper, we refer $G$ to be a simple connected
graph.

######  Definition 1.1.

[10] In a graph $G$, the smallest number $\mu(G,x)$ for $x\in G$ such that
minimum single pebble can be shifted to $x$ using a monophonic path by the
successive pebbling shifts to place $\mu(G,x)$ pebbles on the vertices of $G$.
The maximum $\mu(G,x)$ over $v\in V(G)$ is the monophonic pebbling number
denoted by $\mu(G)$.

######  Definition 1.2.

[8] Any vertex $v\in V(G)$ is called a source vertex when $D(u,v)$ is maximum.

######  Notation 1.1.

In this article we use

a. $D_{vc}$ for the vertex cover set,

b. ${\dot{s}}$ to denote the source vertex,

c. $D_{m}$ for the detour monophonic distance and

d. $\mu_{vc}(G)$ to denote the detour monophonic vertex cover pebbling number
- $DMVCPN$.

##  2\. Motivation

Motivated by the concepts of the cover pebbling number [3] and detour
monophonic number [10] and monophonic pebbling number [6] with vast
applicatoins, we have discussed the detour monophonic vertex cover pebbling
number for some standard graphs such as $C_{n}(n\geq 3),P_{n},F_{n}$ and
$W_{n}$. Detour monophonic vertex cover pebbling is applied in deciding the
equal distribution of goods on every customer by using the detour monophonic
path. This is also applied in the network transmission of the information from
one node to the other.

######  Theorem 2.1.

Let $P_{n}$ be a path of order $n\ (n\ is\ odd\ and\ n\geq 3)$.
Then $\mu_{vc}(P_{n})=\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}$.

###### Proof.

Let $P_{n}$ be a path of order $n$, where $n$ is odd. Let $D_{vc}$ be the
vertex cover set. Then, $D_{vc}=\\{v_{2},v_{4},v_{6},\cdots,v_{n-1}\\}.$ Let
the initial configuration be
$\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}-1$ pebbles. Without loss
of generality let ${\dot{s}}=v_{1}$. The detour monophonic distance from
$v_{1}$ to any other vertex is at most $n-2$. In order to cover the farthest
vertex $v_{n-1}$ using the detour monophonic path we require $2^{n-2}$
pebbles. Similarly, to cover the remaining vertices of $D_{vc}$, we require
the following pebble distributions.
$2^{1}+2^{3}+2^{5}+2^{7}+\cdots+2^{n-2}$. That sums upto
$\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}$. But our initial
configuration of pebbles is
$\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}-1$ which is a
contradiction. Therefore,

$\mu_{vc}(P_{n})\geq\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}$.

Now we prove the sufficient condition by using
$\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}$ pebbles on the vertices
of $P_{n}$.

Case 1: Let ${\dot{s}}=v_{1}\ \text{or}\ v_{n}$.
Without loss of generality, let $v_{n}$ be the source vertex. Then
$D*_{vc}=\\{v_{n-1},v_{n-3},v_{n-5},\cdots,v_{2}\\}$.

| $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{n-5}$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---
$v_{1}$ | $1$ | $3$ | $5$ | $\cdots$ | $n-6$ | $n-4$ | $n-2$
$v_{n}$ | $n-2$ | $n-4$ | $n-6$ | $\cdots$ | $5$ | $3$ | $1$
Table 1. Detour monophonic distance from end vertices to $V(D*_{vc})$.

In order to cover the vertices of $P_{n}$, it suffices to cover the vertices
of vertex cover set $V(D*_{vc})$. From Table 1, we require $2^{n-2}$,
$2^{n-4}$ and $2^{n-6}$ pebbles from $v_{n}$ to cover $v_{2}$, $v_{4}$ and
$v_{6}$ respectively. Proceeding in the same way we use $2^{5},2^{3}$ and $2$
pebbles to cover the vertices $v_{n-5}$, $v_{n-3}$ and $v_{n-1}$. Thus the
following series of pebbles guarantees covering all the vertices of $D*_{vc}$.
$\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{n-5})+\mu_{vc}(G,v_{n-3})+\mu_{vc}(G,v_{n-1})$
$=2^{n-2}+2^{n-4}+2^{n-6}+\cdots+2^{5}+2^{3}+2^{1}$. Thus the number of
pebbles used to cover $V(D*_{vc})$ is,
$\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}$.

Case 2: When the middle vertex is the source vertex.
Let ${\dot{s}=v_{i},i={\lfloor\frac{n}{2}\rfloor}+1}$.
Subcase 2.1: If $n\equiv 1(mod~{}4)$ and then
$D^{1}_{vc}=\\{v_{2},v_{4},v_{6},\cdots$, $v_{\lfloor{\frac{n}{2}\rfloor}-2}$
, $v_{\lfloor{\frac{n}{2}\rfloor}}$ , $v_{\lfloor{\frac{n}{2}\rfloor}+2}$ ,
$v_{\lfloor{\frac{n}{2}\rfloor}+4},\cdots,v_{n-1}\\}.$

| $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{\lfloor{\frac{n}{2}\rfloor}-2}$ | $v_{\lfloor{\frac{n}{2}\rfloor}}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+2}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+4}$ | $\cdots$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{\lfloor{\frac{n}{2}}\rfloor+1}$ | ${\lfloor{\frac{n}{2}\rfloor}-1}$ | ${\lfloor{\frac{n}{2}\rfloor}-3}$ | ${\lfloor{\frac{n}{2}\rfloor}-5}$ | $\cdots$ | $3$ | $1$ | $1$ | $3$ | $\cdots$ | ${\lfloor{\frac{n}{2}\rfloor}-3}$ | ${\lfloor{\frac{n}{2}\rfloor}-1}$
Table 2. Detour monophonic distance from the middle vertex of $P_{n}$ to
$V(D^{1}_{vc})$, where $n\equiv 1(mod~{}4)$ .

From Table 2, we use $2^{\lfloor{\frac{n}{2}\rfloor}-1}$,
$2^{\lfloor{\frac{n}{2}\rfloor}-3}$ and $2^{\lfloor{\frac{n}{2}\rfloor}-5}$
pebbles from $v_{\lfloor\frac{n}{2}\rfloor}+1$ to cover $v_{2}$, $v_{4}$ and
$v_{6}$. We continue in this manner and cover the vertices of $D^{1}_{vc}$
with the following pebble distribution.
$\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}-2})+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}})+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}+2})+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}+4})+\cdots+\mu_{vc}(G,v_{n-3})+\mu_{vc}(G,v_{n-1})$.
$=2^{{\lfloor{\frac{n}{2}\rfloor}-1}}+2^{\lfloor{\frac{n}{2}\rfloor}-3}+2^{\lfloor{\frac{n}{2}\rfloor}-5}+\cdots+2^{3}+2^{1}+2^{1}+2^{3}+\cdots+2^{\lfloor{\frac{n}{2}\rfloor}-3}+2^{\lfloor{\frac{n}{2}\rfloor}-1}$
$=2(2^{1}+2^{3}+2^{5}+2^{7}+\cdots+2^{\lfloor{\frac{n}{2}}\rfloor-1}$).
Thus in order to cover the vertices of $D^{1}_{vc}$ we used
$2\big{(}\sum_{j=1}^{\lfloor{\frac{n}{4}}\rfloor}2^{2(j)-1}\big{)}$ pebbles
which is less than the total number of pebbles available.

Subcase 2.2: Let $n\equiv 3(mod~{}4)$. Without loss of generality
$D^{2}_{vc}=\\{v_{2},v_{4},v_{6},\cdots$ , $v_{\lfloor{\frac{n}{2}\rfloor}-1}$
, $v_{\lfloor{\frac{n}{2}\rfloor}+1}$ ,
$v_{\lfloor{\frac{n}{2}\rfloor}+3},\cdots,v_{n-1}\\}$.

| $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{\lfloor{\frac{n}{2}\rfloor}-1}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+1}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+3}$ | $\cdots$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---
$v_{\lfloor{\frac{n}{2}}\rfloor+1}$ | ${\lfloor{\frac{n}{2}\rfloor}-1}$ | ${\lfloor{\frac{n}{2}\rfloor}-3}$ | ${\lfloor{\frac{n}{2}\rfloor}-5}$ | $\cdots$ | $2$ | $0$ | $2$ | $\cdots$ | ${\lfloor{\frac{n}{2}\rfloor}-3}$ | ${\lfloor{\frac{n}{2}\rfloor}-1}$
Table 3. Detour monophonic distance from the middle vertex of $P_{n}$ to
$V(D^{2}_{vc})$, where $n\equiv 3(mod~{}4)$ .

From Table 3, we use $2^{\lfloor{\frac{n}{2}\rfloor}-1}$,
$2^{\lfloor{\frac{n}{2}\rfloor}-3}$ and $2^{\lfloor{\frac{n}{2}\rfloor}-5}$
pebbles to cover $v_{2}$, $v_{4}$ and $v_{6}$. Similarly we cover the vertices
of $D^{2}_{vc}$ with the following pebble distributions.
$\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}-1})+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}+1})+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}+3})+\cdots+\mu_{vc}(G,v_{n-3})+\mu_{vc}(G,v_{n-1})$.
$=2^{{\lfloor{\frac{n}{2}\rfloor}-1}}+2^{\lfloor{\frac{n}{2}\rfloor}-3}+2^{\lfloor{\frac{n}{2}\rfloor}-5}+\cdots+2^{2}+2^{0}+2^{2}+\cdots+2^{\lfloor{\frac{n}{2}\rfloor}-3}+2^{\lfloor{\frac{n}{2}\rfloor}-1}$
$=2^{0}+2(2^{2}+2^{4}+2^{6}+\cdots+2^{\lfloor{\frac{n}{2}}\rfloor-1})$.

Thus, we used $2\big{(}\sum_{k=1}^{\lfloor{\frac{n}{4}}\rfloor}2^{k}\big{)}+1$
pebbles, which is less than
$\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}$.

Case 3: Consider ${\dot{s}}$ is an internal vertex. Let ${\dot{s}=v_{i}}$,
where $2\leq i\leq n-1$, $i\neq{{\lfloor{\frac{n}{2}}\rfloor}+1}$.

Subcase 3.1: When $i$ is even.
Let $V(D^{3}_{vc})=\\{v_{2}$, $v_{4}$ , $v_{6}$ , $\cdots$ , $v_{i-2}$ ,
$v_{i}$ , $v_{i+2}$ , $\cdots$ , $v_{n-3}$ , $v_{n-1}\\}$.

| $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{i-2}$ | $v_{i}$ | $v_{i+2}$ | $\cdots$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $|i-2|$ | $|i-4|$ | $|i-6|$ | $\cdots$ | ${2}$ | ${0}$ | ${2}$ | $\cdots$ | ${n-(i+3)}$ | ${n-(i+1)}$
Table 4. Detour monophonic distance from the internal vertex of $P_{n}$ to
$V(D^{3}_{vc})$.

When $i=2$, we use $2^{0}$, $2^{2}$, $2^{6},\cdots,2^{n-5}$ and $2^{n-3}$
pebbles to cover $v_{2}$, $v_{4}$, $v_{6},\cdots$, $v_{n-3}$ and $v_{n-5}$
respectively. when $i=4$, we use $2^{2}$, $2^{0}$, $2^{2},\cdots,2^{n-7}$ and
$2^{n-5}$ pebbles to cover $v_{2}$, $v_{4}$, $v_{6},\cdots$, $v_{n-3}$ and
$v_{n-5}$ respectively. Continuing this process, when $i$ is even, we cover
the vertices of $D^{3}_{vc}$ with the following pebble distribution.
$\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{i-2})+\mu_{vc}(G,v_{i})+\mu_{vc}(G,v_{i+2})+\cdots+\mu_{vc}(G,v_{n-3})+\mu_{vc}(G,v_{n-1})$.
From Table 4, we can sum up as follows.
$\sum_{l=1}^{{\frac{i}{2}}-1}2^{2(l)}+\sum_{m=0}^{{\frac{{n}-(i+1)}{2}}}2^{2(m)}$.

Subcase 3.2: When $i$ is odd.
Consider $V(D^{4}_{vc})=\\{v_{2}$, $v_{4}$ , $v_{6}$ , $\cdots$ , $v_{i-3}$ ,
$v_{i-1}$ , $v_{i+1}$ , $v_{i+3}$ , $\cdots$ , $v_{n-3}$ , $v_{n-1}\\}$.

| $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{i-3}$ | $v_{i-1}$ | $v_{i+1}$ | $v_{i+3}$ | $\cdots$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $|i-2|$ | $i-4$ | $i-6$ | $\cdots$ | ${3}$ | ${1}$ | ${1}$ | ${3}$ | $\cdots$ | ${n-(i+3)}$ | ${n-(i+1)}$
Table 5. Detour monophonic distance from the internal vertices of $P_{n}$ to
$V(D^{4}_{vc})$.

From Table 5, when $i$ is odd, we cover the vertices of $D^{4}_{vc}$ with the
following pebble distribution.
$\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{n-3})+\mu_{vc}(G,v_{i-1})+\mu_{vc}(G,v_{i+1})+\mu_{vc}(G,v_{i+3})+\cdots+\mu_{vc}(G,v_{n-3})+\mu_{vc}(G,v_{n-1})$.
$=2^{1}+2^{3}+\cdots+2^{i-4}+2^{i-2}+2^{1}+2^{3}+\cdots+2^{n-(i+3)}+2^{{n-(i+1)}}$.
Thus, we have utilized
$\sum_{l=1}^{\lfloor{\frac{i}{2}}\rfloor}2^{2(l)-1}+\sum_{m=0}^{\lfloor{\frac{{n}-(i+1)}{2}}\rfloor}2^{2(m)+1}$
pebbles, which is less than the total number of pebbles available. Hence,
$\mu_{vc}(P_{n})=\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1}$.
∎

######  Theorem 2.2.

Let $P_{n}$ be a path of order $n(n\ is\ even\ and\ n\geq 2)$.
Then $\mu_{vc}(P_{n})=\sum_{i=0}^{{\frac{n}{2}}-1}2^{2(i)}$.

###### Proof.

Let $P_{n}$ be a path of order $n$, where $n$ is even. Let $D_{vc}$ be the
vertex cover set. Then, $D_{vc}=\\{v_{1},v_{3},v_{5},\cdots,v_{n-1}\\}.$ Let
the initial configuration be $\sum_{i=0}^{{\frac{n}{2}}-1}2^{2(i)}-1$ and
${\dot{s}}=v_{1}$. The detour monophonic distance from $v_{1}$ to any other
vertex of $P_{n}$ is at most $n-2$. Then using $2^{n-2}$ pebbles we could
cover $v_{n-1}$. Similarly to cover $v_{n-3}$, we require $2^{n-4}$ pebbles.
In a similar way we can cover the remaining vertices of $D_{vc}$ using the
following pebble distributions.
$2^{2}+2^{4}+2^{6}+\cdots+2^{n-4}+2^{n-2}$ that sums upto
$\sum_{i=0}^{{\frac{n}{2}}-1}2^{2(i)}-1$. Now, there exists a vertex $v_{1}$
uncovered. Thus we fail to cover all the vertices of $D_{vc}$. Hence,
$\mu_{vc}(P_{n})\geq\sum_{i=0}^{{\frac{n}{2}}-1}2^{2(i)}$.

Now we prove the sufficient condition with
$\sum_{i=0}^{{\frac{n}{2}}-1}2^{2(i)}$ pebbles on the vertices of $P_{n}$.

Case 1:

Assume the end vertex to be the source vertex.
Let ${\dot{s}}=v_{i}$, where $i\in\\{1,n\\}$.
Without loss of generality let ${\dot{s}}=v_{1}$ and
$D^{1}_{vc}=\\{v_{1},v_{3},v_{5},\cdots,v_{n-3},v_{n-1}\\}$.

| $v_{1}$ | $v_{3}$ | $v_{5}$ | $v_{7}$ | $v_{9}$ | $\cdots$ | $v_{n-7}$ | $v_{n-5}$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ | $0$ | $2$ | $4$ | $6$ | $8$ | $\cdots$ | $n-8$ | $n-6$ | $n-4$ | $n-2$
Table 6. Detour monophonic distance from end vertex to $V(D^{1}_{vc})$.

We make use of $2^{0}$, $2^{2}$, $2^{4}$ pebbles to cover the vertices
$v_{1}$, $v_{3}$ and $v_{5}$ respectively. Proceeding in the same way we use
$2^{n-4}$ and $2^{n-2}$ pebbles to cover the vertices $v_{n-3}$ and $v_{n-1}$.
Thus, from Table 6 the following series of pebble distribution guarantees
covering of all the vertices of $D^{1}_{vc}$.
$\mu_{vc}(G,v_{1})+\mu_{vc}(G,v_{3})+\mu_{vc}(G,v_{5})+\mu_{vc}(G,v_{7})+\cdots+\mu_{vc}(G,v_{n-7})+\mu_{vc}(G,v_{n-5})+\mu_{vc}(G,v_{n-3})+\mu_{vc}(G,v_{n-1})$
$=2^{0}+2^{2}+2^{4}+\cdots+2^{n-2}$.
$=\sum_{i=0}^{{\frac{n}{2}}-1}2^{2(i)}$.
By symmetry, we can cover the vertices of $D^{1}_{vc}$, when the source vertex
is $v_{n}$.

Case 2:

Let the middle vertex be the source vertex. Then there exist two subcases.
Subase 2.1:

When $n$ is even and $n\equiv 0(mod~{}4)$.
Assume ${\dot{s}}=v_{i}$.
If $i={\frac{n}{2}}$, then
$D^{2}_{vc}=\\{v_{1},v_{3},v_{5},\cdots,v_{\frac{n}{2}-1},v_{\frac{n}{2}+1},\cdots,v_{n-3},v_{n-1}\\}$.
If $i={\frac{n}{2}+1}$, then
$D^{2}*_{vc}=\\{v_{2},v_{4},v_{6},\cdots,v_{\frac{n}{2}-1},v_{\frac{n}{2}+1},\cdots,v_{n}\\}$.
Without loss of generality, consider the vertex cover set $D^{2}_{vc}$.

| $v_{1}$ | $v_{3}$ | $v_{5}$ | $\cdots$ | $v_{\frac{n}{2}-3}$ | $v_{\frac{n}{2}-1}$ | $v_{\frac{n}{2}+1}$ | $v_{\frac{n}{2}+3}$ | $\cdots$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $\frac{n}{2}-1$ | $\frac{n}{2}-3$ | $\frac{n}{2}-5$ | $\cdots$ | $3$ | $1$ | $1$ | $3$ | $\cdots$ | $\frac{n}{2}-3$ | $\frac{n}{2}-1$
Table 7. Detour monophonic distance from middle vertex to $V(D^{2}_{vc})$.

Now, from Table 7 the following pebble distribution covers the vertices of
$D^{2}_{vc}$.

$2^{{{\frac{n}{2}}-1}}+2^{\frac{n}{2}-3}+2^{\frac{n}{2}-5}+\cdots+2^{3}+2^{1}+2^{1}+2^{3}+\cdots+2^{\frac{n}{2}-3}+2^{\frac{n}{2}-1}$
$=2(2^{1}+2^{3}+2^{5}+\cdots+2^{\frac{n}{2}-1})$.
Similarly, we can cover the vertices of $D^{2*}_{vc}$. Thus in this case to
cover the vertices of the vertex cover set, we use
$2\big{(}\sum_{j=0}^{\frac{n}{4}}2^{2(j)-1}\big{)}$ pebbles, where $n\equiv
0(mod~{}4)$.

Subcase 2.2:

When $n$ is even and $n\equiv 2(mod~{}4)$.
Let $i={\frac{n}{2}}$. Then ${\dot{s}}=v_{\frac{n}{2}}$.
As we consider the odd middle vertex, the vertex cover set also includes all
the odd vertices of $P_{n}$. Thus,
$D^{3}_{vc}=\\{v_{1},v_{3},v_{5},\cdots,v_{\frac{n}{2}},v_{\frac{n}{2}+2},\cdots,v_{n-1}\\}$.
Let $i={\frac{n}{2}+1}$ and ${\dot{s}}=v_{\frac{n}{2}+1}$. When we consider
the even middle vertex, the vertex cover set consists of all the even vertices
of $P_{n}$. Hence
$D^{3*}_{vc}=\\{v_{2},v_{4},v_{6},\cdots,v_{\frac{n}{2}-1},v_{\frac{n}{2}+1},v_{\frac{n}{2}+3},\cdots,v_{n}\\}$.
Without loss of generality, assume the vertex cover set $D^{3*}_{vc}$.

| $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{\frac{n}{2}-3}$ | $v_{\frac{n}{2}-1}$ | $v_{\frac{n}{2}+1}$ | $v_{\frac{n}{2}+3}$ | $v_{\frac{n}{2}+5}$ | $\cdots$ | $v_{n-2}$ | $v_{n}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $\frac{n}{2}-1$ | $\frac{n}{2}-3$ | $\frac{n}{2}-5$ | $\cdots$ | $4$ | $2$ | $0$ | $2$ | $4$ | $\cdots$ | $\frac{n}{2}-3$ | $\frac{n}{2}-1$
Table 8. Detour monophonic distance from middle vertex to $V(D^{3*}_{vc})$.

Thus, from Table 8 we cover the vertices of $D^{3*}_{vc}$ with the following
pebble distribution.
$2^{{\frac{n}{2}}-1}+2^{{\frac{n}{2}}-3}+2^{{\frac{n}{2}}-5}+\cdots+2^{4}+2^{2}+2^{0}+2^{2}+2^{4}+\cdots+2^{{\frac{n}{2}}-3}+2^{{\frac{n}{2}}-1}$.
$=2^{2{\lfloor{\frac{n}{4}\rfloor}}}+\cdots+2^{4}+2^{2}+2^{0}+2^{2}+2^{4}+\cdots+2^{2{\lfloor{\frac{n}{4}\rfloor}}}$.
In this manner we can cover the vertices of $D^{3}_{vc}$. Hence, the pebble
distribution is summed up as follows.
$1+2\big{(}\sum_{k=1}^{\lfloor{\frac{n}{4}\rfloor}}2^{2(k)}\big{)},$ where
$n\equiv 2(mod~{}4)\text{\ and }{\dot{s}}$ is a middle vertex.

Case 3: Consider ${\dot{s}}$ to be an internal vertex. Let ${\dot{s}=v_{i}}$,
where $2\leq i\leq n-1$, $i\neq{\frac{n}{2}},{{\frac{n}{2}}+1}$.

Subase 3.1:

If $v_{i}$ is even and $i<{\frac{n}{2}}$. Then
$D^{4}_{vc}=\\{v_{1},v_{3},v_{5},\cdots,v_{i-1},v_{i+1},\cdots,v_{n-1}\\}$.

| $v_{1}$ | $v_{3}$ | $v_{5}$ | $\cdots$ | $v_{i-3}$ | $v_{i-1}$ | $v_{i+1}$ | $v_{i+3}$ | $\cdots$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $i-1$ | $i-3$ | $i-5$ | $\cdots$ | $3$ | $1$ | $1$ | $3$ | $\cdots$ | $n-(i+1)$
Table 9. Detour monophonic distance from $v_{i}$ to $V(D^{4}_{vc})$.

From Table 9 to cover the vertices of $D^{4}_{vc}$, we require the following
pebble series. The vertices $v_{1}$ and $v_{n-1}$ are covered with $2^{i-1}$
and $2^{n-(i+1)}$ pebbles. Thus the required pebble distribution results as
follows.
$\mu_{vc}(G,v_{1})+\mu_{vc}(G,v_{3})+\cdots+\mu_{vc}(G,v_{i-3})+\mu_{vc}(G,v_{i-1})+\mu_{vc}(G,v_{i+1})+\mu_{vc}(G,v_{i+3})+\cdots+\mu_{vc}(G,v_{n-1})$.
$=2^{i-1}+2^{i-3}+\cdots+2^{3}+2^{1}+2^{1}+2^{3}+\cdots+2^{n-(i+1)}$
$=\sum_{j=1}^{\lfloor{\frac{i}{2}}\rfloor}2^{2(j)-1}+\sum_{k=1}^{\lfloor{\frac{n-(i+1)}{2}}\rfloor}2^{2(k)-1}$.

When $i>{\frac{n}{2}+1}$, we obtain the reversal result using the similar
pebble distribution. Hence the number of pebbles employed in the process of
covering $D^{4}_{vc}$ is,
$\sum_{j=1}^{\lfloor{\frac{n-(i+1)}{2}}\rfloor}2^{2(j)-1}+\sum_{k=1}^{\lfloor{\frac{i}{2}}\rfloor}2^{2(k)-1}$.

Subcase 3.2:

If $v_{i}$ is odd and $i<{\frac{n}{2}}$. Then
$D^{5}_{vc}=\\{v_{1},v_{3},v_{5},\cdots,v_{i},v_{i+2},\cdots,v_{n-1}\\}$.

| $v_{1}$ | $v_{3}$ | $v_{5}$ | $\cdots$ | $v_{i-4}$ | $v_{i-2}$ | $v_{i}$ | $v_{i+2}$ | $v_{i+4}$ | $\cdots$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $i-1$ | $i-3$ | $i-5$ | $\cdots$ | $4$ | $2$ | $0$ | $2$ | $4$ | $\cdots$ | $n-(i+1)$
Table 10. Detour monophonic distance from $v_{i}$ to $V(D^{5}_{vc})$.

The vertex $v_{1}$ is covered with $2^{i-1}$ pebbles and $v_{n-1}$ is covered
with $2^{n-(i+1)}$ pebbles. Thus, from Table 10 to cover the vertices of
$D^{5}_{vc}$, we require the following pebble distributions.
$\mu_{vc}(G,v_{1})+\cdots+\mu_{vc}(G,v_{i-4})+\mu_{vc}(G,v_{i-2})+\mu_{vc}(G,v_{i})+\mu_{vc}(G,v_{i+2})+\mu_{vc}(G,v_{i+4})+\cdots+\mu_{vc}(G,v_{n-1})$.
$=2^{i-1}+\cdots+2^{4}+2^{2}+2^{0}+2^{2}+2^{4}+\cdots+2^{n-(i+1)}$
$=\sum_{l=0}^{\lfloor{\frac{i}{2}}\rfloor}2^{2(l)}+\sum_{m=1}^{\lfloor{\frac{n-(i+1)}{2}}\rfloor}2^{2(m)}$.

If $i>{\frac{n}{2}+1}$, we obtain the reversal result using the similar pebble
distribution. Hence,
$\sum_{l=1}^{\lfloor{\frac{n-(i+1)}{2}}\rfloor}2^{2(l)}+\sum_{m=0}^{\lfloor{\frac{i}{2}}\rfloor}2^{2(m)}$.
Thus in each case, we have used the pebbles which sumps up less than the
considered pebbles. Therefore,
$\mu_{vc}(P_{n})=\sum_{i=0}^{{\frac{n}{2}}-1}2^{2(i)}$.
∎

######  Theorem 2.3.

For Fan $F_{n}(n\ is\ even),\
{\mu_{vc}}(F_{n})=\sum_{i=0}^{{\frac{n}{2}}-2}2^{2(i)+1}+2$.

###### Proof.

Let $F_{n}=P_{n-1}+v_{0}$ be a fan of order $n(n$ is even) with
$V(F_{n})=\\{v_{0},v_{1},v_{2},\cdots,v_{n-1}\\}$.
Let $D_{vc}$ be the vertex cover set. Then,
$D_{vc}=\\{v_{0},v_{2},v_{4},v_{6},\cdots,v_{n-2}\\}.$ Consider the following
configuration of pebbles on the vertices of fan $F_{n}$.
Let the initial configuration be $\sum_{i=0}^{{\frac{n}{2}}-2}2^{2(i)+1}+1$
pebbles. Assume $v_{1}$ to be the source vertex. The detour monophonic
distance from $v_{1}$ to any other vertex is at most $n-2$. Then using
$2^{n-3}$ pebbles we could place a pebble on $v_{n-2}$. Then we place
$2^{n-5}$ pebbles on $v_{n-4}$. In a similar way, we distribute the pebbles on
the vertices of $D_{vc}$. Now, there remains only one pebble on $v_{2}$
leaving $v_{1}$ pebbleless. Thus, we fail to cover all the vertices of
$D_{vc}$. Therefore,
$\mu_{vc}(F_{n})\geq\sum_{i=0}^{{\frac{n}{2}}-2}2^{2(i)+1}+2$.
Now we prove the sufficient condition by using
$\sum_{i=0}^{{\frac{n}{2}}-2}2^{2(i)+1}+2$ pebbles on the vertices of $F_{n}$.

Case 1: Let ${\dot{s}}=v_{1}\ \text{or}\ v_{n-1}$.
Without loss of generality, let $v_{n}$ be the source vertex. Then
$D^{1}_{vc}=\\{v_{0},v_{2},v_{4},v_{6},\cdots,\\\ v_{n-6},v_{n-4},v_{n-2}\\}$.

| $v_{0}$ | $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{n-6}$ | $v_{n-4}$ | $v_{n-2}$
---|---|---|---|---|---|---|---|---
$v_{1}$ | $1$ | $1$ | $3$ | $5$ | $\cdots$ | $n-7$ | $n-5$ | $n-3$
$v_{n-1}$ | $1$ | $n-3$ | $n-5$ | $n-7$ | $\cdots$ | $5$ | $3$ | $1$
Table 11. Detour monophonic distance from $v_{1}$ and $v_{n-1}\ to\
V(D^{1}_{vc})$.

From Table 11, we use $2$ pebbles to cover $v_{0}$. Also, we ues $2^{n-3}$,
$2^{n-5}$, $2^{n-7}$ pebbles to cover $v_{0}$, $v_{2}$, $v_{4}$ and $v_{6}$
respectively. Proceeding in the same way we need $2^{5},2^{3}$ and $2$ pebbles
to cover the vertices $v_{n-6}$, $v_{n-4}$ and $v_{n-2}$. Thus, the following
series of pebbles guarantees covering all the vertices of $D^{1}_{vc}$.
$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{n-6})+\mu_{vc}(G,v_{n-4})+\mu_{vc}(G,v_{n-2})$
$=2^{1}+2^{n-3}+2^{n-5}+2^{n-7}+\cdots+2^{5}+2^{3}+2^{1}$. Thus,
$\sum_{i=0}^{{\frac{n}{2}}-2}2^{2(i)+1}+2$ pebbles are used to cover
$D^{1}_{vc}$.

Case 2:

Consider ${\dot{s}=v_{i},i={\frac{n}{2}}}$.
Subcase 2.1: If $n\equiv 2(mod~{}4)$ and then
$D^{2}_{vc}=\\{v_{0},v_{2},v_{4},v_{6},\cdots,v_{\frac{n}{2}-3},v_{\frac{n}{2}-1},v_{\frac{n}{2}+1},\\\
v_{\frac{n}{2}+3},\cdots,v_{n-4},v_{n-2}\\}.$

| $v_{0}$ | $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{\frac{n}{2}-3}$ | $v_{\frac{n}{2}-1}$ | $v_{\frac{n}{2}+1}$ | $v_{\frac{n}{2}+3}$ | $\cdots$ | $v_{n-4}$ | $v_{n-2}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{\frac{n}{2}}$ | $1$ | ${\frac{n}{2}-2}$ | ${\frac{n}{2}-4}$ | ${\frac{n}{2}-6}$ | $\cdots$ | $3$ | $1$ | $1$ | $3$ | $\cdots$ | ${\frac{n}{2}-4}$ | ${\frac{n}{2}-2}$
Table 12. Detour monophonic distance from $v_{\frac{n}{2}}$ of $F_{n}$ to
$V(D^{2}_{vc})$, where $n\equiv 2(mod~{}4)$ .

From Table 12, we use $2^{\frac{n}{2}-2}$, $2^{\frac{n}{2}-4}$ pebbles to
cover $v_{2}$, $v_{n-2}$ and $v_{4}$, $v_{n-4}$ respectively. Obviously.
$v_{0}$ is covered with 2 pebbles. We continue in this manner and cover the
vertices of $D^{2}_{vc}$ with the following pebble distribution.

$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{\frac{n}{2}-3})\mu_{vc}(G,v_{\frac{n}{2}-1})+\mu_{vc}(G,v_{\frac{n}{2}+1})+\mu_{vc}(G,v_{\frac{n}{2}+3})+\cdots+\mu_{vc}(G,v_{n-4})+\mu_{vc}(G,v_{n-2})$.

$=2^{1}+2^{\frac{n}{2}-2}+2^{\frac{n}{2}-4}+2^{\frac{n}{2}-6}+\cdots+2^{3}+2^{1}+2^{1}+2^{3}+\cdots+2^{\frac{n}{2}-4}+2^{\frac{n}{2}-2}$.
Thus in order to cover the vertices of $D^{2}_{vc}$ we used
$2\big{(}\sum_{j=1}^{\lfloor{\frac{n}{4}}\rfloor}2^{2(j)-1}\big{)}+2$ pebbles,
which is less than the total number of pebbles available.

Subcase 2.2: Let $n\equiv 0(mod~{}4)$. Without loss of generality
$D^{3}_{vc}=\\{v_{0},v_{2},v_{4},v_{6},\cdots,\\\
v_{\frac{n}{2}-2},v_{\frac{n}{2}},v_{\frac{n}{2}+2},\cdots,v_{n-4},v_{n-2}\\}$.

| $v_{0}$ | $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{\frac{n}{2}-2}$ | $v_{\frac{n}{2}}$ | $v_{\frac{n}{2}+2}$ | $\cdots$ | $v_{n-4}$ | $v_{n-2}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{\frac{n}{2}}$ | $1$ | ${\frac{n}{2}-2}$ | ${\frac{n}{2}-4}$ | ${\frac{n}{2}-6}$ | $\cdots$ | $2$ | $0$ | $2$ | $\cdots$ | ${\frac{n}{2}-4}$ | ${\frac{n}{2}-2}$
Table 13. Detour monophonic distance from the middle vertex of $F_{n}$ to
$V(D^{3}_{vc})$, where $n\equiv 0(mod~{}4)$ .

From Table 13, $v_{0}$ is covered with 2 pebbles. We use $2^{\frac{n}{2}-2}$,
$2^{\frac{n}{2}-4}$ and $2^{\frac{n}{2}-6}$ pebbles to cover $v_{2}$, $v_{4}$
and $v_{6}$. Similarly we cover the vertices of $D^{3}_{vc}$ with the
following pebble distributions.
$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{\frac{n}{2}-2})+\mu_{vc}(G,v_{\frac{n}{2}})+\mu_{vc}(G,v_{\frac{n}{2}+2})+\cdots+\mu_{vc}(G,v_{n-4})+\mu_{vc}(G,v_{n-2})$.
$=2^{0}+2(2^{2}+2^{4}+2^{6}+\cdots+2^{{\frac{n}{4}}-1})+2^{1}$.

Thus we used $2\big{(}\sum_{k=1}^{{{\frac{n}{4}}}-1}2^{k}\big{)}+3$, which is
less than $\sum_{i=0}^{{\frac{n}{2}}-2}2^{2(i)+1}+2$.

Case 3: Consider ${\dot{s}}$ is an internal vertex. Let ${\dot{s}=v_{i}}$,
where $2\leq i\leq n-2$, $i\neq{{\lfloor{\frac{n-1}{2}}\rfloor}+1}$.

Subcase 3.1: When $i$ is even.
Let $V(D^{4}_{vc})=\\{v_{0},v_{2}$, $v_{4}$ , $v_{6}$ , $\cdots$ , $v_{i-2}$ ,
$v_{i}$ , $v_{i+2}$ , $\cdots$ , $v_{n-4}$ , $v_{n-2}\\}$.

| $v_{0}$ | $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{i-2}$ | $v_{i}$ | $v_{i+2}$ | $\cdots$ | $v_{n-4}$ | $v_{n-2}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | ${1}$ | $|i-2|$ | $|i-4|$ | $|i-6|$ | $\cdots$ | ${2}$ | ${0}$ | ${2}$ | $\cdots$ | ${n-(i+4)}$ | ${n-(i+2)}$
Table 14. Detour monophonic distance from $v_{i}$ of $F_{n}$ to
$V(D^{4}_{vc})$.

From Table 14, when $i=2$, we use 2, $2^{0}$, $2^{2}$, $2^{6},\cdots,2^{n-6}$
and $2^{n-4}$ pebbles to cover $v_{0}$, $v_{2}$, $v_{4}$, $v_{6},\cdots$,
$v_{n-4}$ and $v_{n-6}$ respectively. when $i=4$, we use $2^{0}$, $2^{2}$,
$2^{0}$, $2^{2},\cdots,2^{n-8}$ and $2^{n-5}$ pebbles to cover $v_{2}$,
$v_{4}$, $v_{6},\cdots$, $v_{n-4}$ and $v_{n-6}$ respectively. Continuing this
process, when $i$ is even, we cover the vertices of $D^{4}_{vc}$ with the
following pebble distribution.
$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{i-2})+\mu_{vc}(G,v_{i})+\mu_{vc}(G,v_{i+2})+\cdots+\mu_{vc}(G,v_{n-4})+\mu_{vc}(G,v_{n-2})$.
$=\sum_{l=1}^{{\frac{i}{2}}-1}2^{2(l)}+\sum_{m=0}^{{\frac{{n}-(i+2)}{2}}}2^{2(m)}+2$.

Subcase 3.2: When $i$ is odd.
Consider $V(D^{5}_{vc})=\\{v_{0},v_{2}$, $v_{4}$ , $v_{6}$ , $\cdots$ ,
$v_{i-3}$ , $v_{i-1}$ , $v_{i+1}$ , $v_{i+3}$ , $\cdots$ , $v_{n-4}$ ,
$v_{n-2}\\}$.

| $v_{0}$ | $v_{2}$ | $v_{4}$ | $v_{6}$ | $\cdots$ | $v_{i-3}$ | $v_{i-1}$ | $v_{i+1}$ | $v_{i+3}$ | $\cdots$ | $v_{n-4}$ | $v_{n-2}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $1$ | $|i-2|$ | $|i-4|$ | $|i-6|$ | $\cdots$ | ${3}$ | ${1}$ | ${1}$ | ${3}$ | $\cdots$ | ${n-(i+4)}$ | ${n-(i+2)}$
Table 15. Detour monophonic distance from $v_{i}$ of $F_{n}$ to
$V(D^{5}_{vc})$.

When $i$ is odd, we cover the vertices of $D^{5}_{vc}$ with the following
pebble distribution.
$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{i-1})+\mu_{vc}(G,v_{i-3})+\mu_{vc}(G,v_{4})+\mu_{vc}(G,v_{6})+\cdots+\mu_{vc}(G,v_{n-6})+\mu_{vc}(G,v_{i-1})+\mu_{vc}(G,v_{i+1})+\mu_{vc}(G,v_{i+3})+\cdots+\mu_{vc}(G,v_{n-4})+\mu_{vc}(G,v_{n-2})$.
$=2^{1}+2^{1}+2^{3}+\cdots+2^{i-4}+2^{i-2}+2^{1}+2^{3}+\cdots+2^{n-(i+4)}+2^{{n-(i+2)}}$.
Thus, we have ended up with
$\sum_{l=1}^{\lfloor{\frac{i}{2}}\rfloor}2^{2(l)-1}+\sum_{m=0}^{\lfloor{\frac{{n}-(i+2)}{2}}\rfloor}2^{2(m)+1}+2$
pebbles which is less than the total number of pebbles available.

Case 4:

Consider the hub vertex as the source vertex. Then ${\dot{s}=v_{0}}$ and
$D^{6}_{vc}=\\{v_{n-2},v_{n-4},v_{n-6},\cdots,v_{2},v_{0}\\}$.
The detour monophonic distance from any other vertex of $F_{n}$ to $v_{0}$ is
1. Thus in order to cover the vertices of $D^{6}_{vc}$, we have the pebble
distribution that sums up as follows.
$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{2})+\mu_{vc}(G,v_{4})+\cdots+\mu_{vc}(G,v_{n-6})+\mu_{vc}(G,v_{n-4})+\mu_{vc}(G,v_{n-2})$
$=2^{0}+2^{1}+2^{1}+\cdots+2^{1}+2^{1}+2^{1}$
$2{\lfloor{\frac{n-1}{2}}\rfloor}+1$ which is less than the considered number
of pebbles.
Hence,
$\mu_{vc}(F_{n})=\sum_{i=0}^{{\frac{n}{2}}-2}2^{2(i)+1}+2$, when $n$ is even.

∎

######  Theorem 2.4.

For Fan $F_{n}(n\ is\
odd),\mu_{vc}(F_{n})=\sum_{j=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(j)}+2$.

###### Proof.

The proof is similar to that of Theorems 2.2 and 2.3. ∎

######  Theorem 2.5.

For Cycle $C_{n}(n\geq 3),\\\ {\mu_{vc}}(C_{n})$ =
$\begin{cases}2\big{(}\sum_{i=\frac{n}{4}+1}^{\frac{n}{2}-1}2^{2(i)}\big{)}+2^{\frac{n}{2}}+1,&\text{if
}n\ is\ \text{even }\text{and }n\equiv 0(mod~{}4)\\\
2\big{(}\sum_{j={\lfloor\frac{n}{4}\rfloor}+1}^{\frac{n}{2}-1}2^{2(j)}\big{)}+1,&\text{if
}n\ is\ \text{even }\text{and }\ n\equiv 2(mod~{}4)\\\
{\sum_{k={\lfloor\frac{n}{2}\rfloor}+1}^{n-2}2^{k}}+3,&\text{if }n\ is\
\text{odd }.\par\end{cases}$

###### Proof.

Let $C_{n}$ be an even cycle of order $n(n\geq 3)$.

Case 1:

When $n\equiv 0(mod~{}4)$.
Consider the detour monophonic vertex cover set
$D^{1}_{vc}=\\{v_{i+2(j)}|i=1,2,3,\cdots,n;j=0,1,2,\cdots,{\frac{n}{4}-1},{\frac{n}{4}},{\frac{n}{4}+1},\cdots,{\frac{n}{2}-2},{\frac{n}{2}-1}\\},$
the indices run over modulus $n$. Thus $|D^{1}_{vc}|=\frac{n}{2}$.

Let ${\dot{s}}=v_{1}$ and the initial configuration be
$2\big{(}\sum_{i=\frac{n}{4}+1}^{\frac{n}{2}-1}2^{2i}\big{)}+2^{\frac{n}{2}}$.
In order to cover the vertices $v_{(n-1)}$ we require $2^{n-2}$ pebbles and to
cover $v_{(n-3)}$ we require $2^{n-4}$ pebbles respectively using the detour
monophonic path. Similar process can be carried on to cover the remaining
vertices of $D^{1}_{vc}$ using the following series of pebble distribution
that sums up
$2^{n-2}+2^{n-4}+\cdots+2^{\frac{n}{2}+2}+2^{\frac{n}{2}}+2^{\frac{n}{2}+2}+\cdots+2^{n-2}+2^{n-4}$
$=2(2^{n-2}+2^{n-4}+\cdots+2^{{\frac{n}{2}+2}})+2^{\frac{n}{2}}$
$=2\big{(}\sum_{i=\frac{n}{4}+1}^{\frac{n}{2}-1}2^{2i}\big{)}+2^{\frac{n}{2}}$.
Thus we can cover the vertices $v_{3}$ and $v_{5}$ with $2^{n-2}$ and
$2^{n-4}$ pebbles and we are left with no pebble to cover the source vertex
$v_{1}$. Therefore, the detour monophonic vertex cover number
${\mu_{vc}}(C_{n})\geq
2\big{(}\sum_{i=\frac{n}{4}+1}^{\frac{n}{2}-1}2^{2(i)}\big{)}+2^{\frac{n}{2}}+1$.

| $v_{i}$ | $v_{i+2}$ | $v_{i+4}$ | $\cdots$ | $v_{{\frac{n}{2}+i-2}}$ | $v_{\frac{n}{2}+i}$ | $v_{{\frac{n}{2}+i+2}}$ | $\cdots$ | $v_{n+i-4}$ | $v_{n+i-2}$
---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $0$ | $n-2$ | $n-4$ | $\cdots$ | $\frac{n}{2}+2$ | $\frac{n}{2}$ | $\frac{n}{2}+2$ | $\cdots$ | $n-4$ | $n-2$
Table 16. Detour monophonic distance from $v_{i}\ to\ V(D^{1}_{vc})$.

Obviously $v_{i}$ is covered with one pebble. From Table 16, in order to cover
the vertices $v_{i+2}$ and $v_{n+i-2}$ we require $2^{n-2}$ pebbles each and
to cover $v_{i+4}$ and $v_{n+i-4}$ we require $2^{n-4}$ pebbles each.
Similarly to cover $v_{{\frac{n}{2}+i-2}}$ and $v_{{\frac{n}{2}+i+2}}$, we
require $2^{\frac{n}{2}+2}$ pebbles. We need $2^{\frac{n}{2}}$ pebbles to
cover the vertex $v_{i+{\frac{n}{2}}}$. Thus in order to cover the vertices in
$D^{1}_{vc}$ we distribute the pebbles as follows.
$\mu_{vc}(G,v_{i})+\mu_{vc}(G,v_{i+2})+\mu_{vc}(G,v_{i+4})+\cdots+\mu_{vc}(G,v_{{\frac{n}{2}+i-2}})+\mu_{vc}(G,v_{{\frac{n}{2}+i}})+\mu_{vc}(G,v_{{\frac{n}{2}+i+2}})+\cdots+\mu_{vc}(G,v_{n+i-4})+\mu_{vc}(G,v_{n+i-2})$.
$=2^{0}+2^{n-2}+2^{n-4}+\cdots+2^{\frac{n}{2}+2}+2^{\frac{n}{2}}+2^{\frac{n}{2}+2}+\cdots+2^{n-4}+2^{n-2}$
$=1+2(2^{n-2}+2^{n-4}+\cdots+2^{{\frac{n}{2}+2}})+2^{\frac{n}{2}}$
$=2\big{(}\sum_{i=\frac{n}{4}+1}^{\frac{n}{2}-1}2^{2i}\big{)}+2^{\frac{n}{2}}+1$.
Since all the vertices are of same detour monophonic distance, the number of
pebbles required to place on any $v_{i}\in V(D^{1}{vc})$ is
$2\big{(}\sum_{i=\frac{n}{4}+1}^{\frac{n}{2}-1}2^{2(i)}\big{)}+2^{\frac{n}{2}}+1$.
Thus,
$\mu_{vc}(C_{n})=2\big{(}\sum_{i=\frac{n}{4}+1}^{\frac{n}{2}-1}2^{2i}\big{)}+2^{\frac{n}{2}}+1$.

Case 2:

When $n$ is even and $n\equiv 2(mod~{}4)$, let
$D^{2}_{vc}=\\{v_{i+2j}|i=1,2,3,\cdots,n;j=0,1,2,\cdots,\lfloor{\frac{n}{4}-1}\rfloor,\lfloor{\frac{n}{4}}\rfloor,\lfloor{\frac{n}{4}+1}\rfloor,\cdots,{\frac{n}{2}-2},{\frac{n}{2}-1}\\}$
be the vertex cover set, where the indices run over modulus $n$. Thus,
$|D^{2}_{vc}|=\frac{n}{2}$. Now consider the configuration which encages
$2\big{(}\sum_{j={\lfloor{\frac{n}{4}}\rfloor}+1}^{\frac{n}{2}-1}2^{2(j)}\big{)}$
pebbles to cover the vertices of $C_{n}$. To cover the vertices $v_{i+2}$ and
$v_{i+n-2}$, we need $2^{n-2}$ pebbles. Similarly to cover the vertices
$v_{i+4}$ and $v_{i+n-4}$, we need $2^{n-4}$ pebbles. We use
$2^{\frac{n}{2}+1}$ pebbles to cover $v_{i+2(\lfloor{\frac{n}{4}}\rfloor)}$
and $v_{i+{2(\lfloor{\frac{n}{4}}\rfloor+1)}}$. Obviously, we require a pebble
to cover ${\dot{s}}$. Thus, $\mu_{vc}(C_{n})\geq
2\big{(}\sum_{j={\lfloor{\frac{n}{4}}\rfloor}+1}^{\frac{n}{2}-1}2^{2(j)}\big{)}+1$.

| $v_{i}$ | $v_{i+2}$ | $v_{i+4}$ | $\cdots$ | $v_{{\frac{n}{2}+i-3}}$ | $v_{{\frac{n}{2}+i-1}}$ | $v_{\frac{n}{2}+i+1}$ | $v_{{\frac{n}{2}+i+3}}$ | $\cdots$ | $v_{n+i-4}$ | $v_{n+i-2}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $0$ | $n-2$ | $n-4$ | $\cdots$ | $\frac{n}{2}+3$ | $\frac{n}{2}+1$ | $\frac{n}{2}+1$ | $\frac{n}{2}+3$ | $\cdots$ | $n-4$ | $n-2$
Table 17. Detour monophonic distance from $v_{i}\ to\ V(D^{2}_{vc})$.

Clearly only one pebble suffices to cover the vertex
$v_{i}(i=1,2,3,\cdots,n)$. From Table 17, all the vertices of $D^{2}_{vc}$ are
covered by the following series of pebble distribution.

$\mu_{vc}(G,v_{i})+\mu_{vc}(G,v_{i+2})+\mu_{vc}(G,v_{i+4})+\cdots+\mu_{vc}(G,v_{{\frac{n}{2}+i-3}})+\mu_{vc}(G,v_{{\frac{n}{2}+i-1}})+\mu_{vc}(G,v_{{\frac{n}{2}+i+1}})+\mu_{vc}(G,v_{{\frac{n}{2}+i+3}})+\cdots+\mu_{vc}(G,v_{n+i-4})+\mu_{vc}(G,v_{n+i-2})$.
$=2^{0}+2^{n-2}+2^{n-4}+\cdots+2^{\frac{n}{2}+1}+2^{\frac{n}{2}+1}+\cdots+2^{n-2}+2^{n-4}\\\
1+2(2^{n-2}+2^{n-4}+\cdots+2^{2({\frac{n}{2}-1})}+2^{2({\frac{n}{2}-2})}+\cdots+2^{2(\lfloor{\frac{n}{4}}\rfloor+1)})$
Thus,
$2\big{(}\sum_{j={\lfloor{\frac{n}{4}}\rfloor}+1}^{\frac{n}{2}-1}2^{2(j)}\big{)}+1$.
Since all the vertices of $C_{n}$ are at equivi distance, the result holds for
any $v_{i}\in V(C_{n})$, when $n$ is even and $n\equiv 2(mod~{}4)$. Hence,
$\mu_{vc}(C_{n})=2\big{(}\sum_{j={\lfloor{\frac{n}{4}}\rfloor}+1}^{\frac{n}{2}-1}2^{2(j)}\big{)}+1$.

Case 3: When $n$ is odd, assume the detour monophonic cover set
$D^{3}_{vc}=\\{v_{i+j}|i=1,2,3,\cdots,n;j=0,1,2,\cdots,n-1\\}$, where all the
indices run over modulus $n$ and $|D_{vc}|={\lfloor\frac{n}{2}\rfloor}+1$.
Without loss of generality we fix ${\dot{s}}=v_{1}$.

Let the initial configuration be
${\sum_{k={\lfloor\frac{n}{2}\rfloor}+1}^{n-2}2^{k}}+2$. The vertices $v_{3}$,
$v_{5}$ and $v_{7}$ are covered with $2^{n-2}$, $2^{n-4}$ and $2^{n-5}$
pebbles respectively. To cover the vertices
$v_{{\lfloor{\frac{n}{2}}\rfloor}-2}$, $v_{{\lfloor{\frac{n}{2}}\rfloor}}$ and
$v_{{\lfloor{\frac{n}{2}}\rfloor}+2}$, we have to use
$2^{\lfloor{\frac{n}{2}\rfloor}+4}$, $2^{\lfloor{\frac{n}{2}\rfloor}+2}$ and
$2^{{\lfloor{\frac{n}{2}\rfloor}+1}}$ pebbles on the corresponding vertices.
Continuing in this manner we are left with $v_{1}$ uncovered. Hence, the
monophonic vertex cover pebbling number

$\mu_{vc}(C_{n})\geq{\sum_{k={\lfloor\frac{n}{2}\rfloor}+1}^{n-2}2^{k}}+3$.
If $n$ is odd and $n\equiv 1(mod~{}4)$, then
$D^{3}_{vc}=\\{v_{1},v_{3},v_{5},\cdots,v_{\lfloor{\frac{n}{2}\rfloor}-1},v_{\lfloor{\frac{n}{2}\rfloor}+1},v_{\lfloor{\frac{n}{2}\rfloor}+3},v_{\lfloor{\frac{n}{2}\rfloor}+5},\cdots,\\\
v_{n-4},v_{n-2},v_{n}\\}$. Without loss of generality consider $v_{1}$ to be
the source vertex.

| $v_{1}$ | $v_{3}$ | $v_{5}$ | $\cdots$ | $v_{\lfloor{\frac{n}{2}\rfloor}-1}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+1}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+3}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+5}$ | $\cdots$ | $v_{n-4}$ | $v_{n-2}$ | $v_{n}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ | $0$ | $n-2$ | $n-4$ | $\cdots$ | $\lfloor{\frac{n}{2}\rfloor}+3$ | ${\lfloor{\frac{n}{2}\rfloor}+1}$ | ${\lfloor{\frac{n}{2}\rfloor}+2}$ | ${\lfloor{\frac{n}{2}\rfloor}+4}$ | $\cdots$ | $n-5$ | $n-3$ | $1$
Table 18. Detour monophonic distance from $v_{1}\ to\ V(D^{3}_{vc})$.

If $n$ is odd and $n\equiv 3(mod~{}4)$, then
$D^{3*}_{vc}=\\{v_{1},v_{3},v_{5},\cdots,v_{\lfloor{\frac{n}{2}\rfloor}-2},v_{\lfloor{\frac{n}{2}\rfloor}},v_{\lfloor{\frac{n}{2}\rfloor}+2},v_{\lfloor{\frac{n}{2}\rfloor}+4},\cdots,\\\
v_{n-4},v_{n-2},v_{n}\\}$. Without loss of generality let $v_{1}$ be the
source vertex.

| $v_{1}$ | $v_{3}$ | $v_{5}$ | $\cdots$ | $v_{\lfloor{\frac{n}{2}\rfloor}-2}$ | $v_{\lfloor{\frac{n}{2}\rfloor}}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+2}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+4}$ | $\cdots$ | $v_{n-4}$ | $v_{n-2}$ | $v_{n}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ | $0$ | $n-2$ | $n-4$ | $\cdots$ | $\lfloor{\frac{n}{2}\rfloor}+4$ | ${\lfloor{\frac{n}{2}\rfloor}+2}$ | ${\lfloor{\frac{n}{2}\rfloor}+1}$ | ${\lfloor{\frac{n}{2}\rfloor}+3}$ | $\cdots$ | $n-5$ | $n-3$ | $1$
Table 19. Detour monophonic distance from $v_{1}\ to\ V(D^{3*}_{vc})$.

In order to cover the vertices of $C_{n}$, it suffices to cover the vertices
of vertex cover set. From Tables 18 and 19, the pebbles are distributed sums
up as follows.

$=2^{0}+2^{1}+2^{\lfloor{\frac{n}{2}\rfloor}+1}+2^{\lfloor{\frac{n}{2}\rfloor}+2}+2^{\lfloor{\frac{n}{2}\rfloor}+3}+2^{\lfloor{\frac{n}{2}\rfloor}+4}+\cdots+2^{n-5}+2^{n-4}+2^{n-3}+2^{n-2}$
$={\sum_{k={\lfloor\frac{n}{2}\rfloor}+1}^{n-2}2^{k}}+3$.
Thus, $\mu_{vc}(C_{n})={\sum_{k={\lfloor\frac{n}{2}\rfloor}+1}^{n-2}2^{k}}+3$.
Since the vertices of $C_{n}$ are equivi distanced, the result is true for any
$v_{i}\in\\{v_{i}|2\leq i\leq n\\}$, when $n$ is odd.

Hence proved.
∎

######  Theorem 2.6.

For a Wheel $W_{n}$,
${\mu_{vc}}(W_{n})$ =
$\begin{cases}{\sum_{i=\frac{n}{2}}^{n-3}2^{i}}+5,&\text{if }n\ is\ \text{even
}\\\
2\big{(}\sum_{j=\lfloor{\frac{n}{4}\rfloor}+1}^{{\lfloor{\frac{n}{2}}\rfloor}-1}2^{2(j)}\big{)}+2^{\lfloor{\frac{n}{2}}\rfloor}+3,&\text{if
}n\ is\ \text{odd }\ \text{and }n\equiv 1(mod~{}4)\\\
2\big{(}\sum_{k={\lfloor{\frac{n}{4}}\rfloor}+1}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(k)}\big{)}+3,&\text{if
}n\ is\ \text{odd}\ \text{and }\ n\equiv 3(mod~{}4)\\\ \end{cases}$

###### Proof.

Let $W_{n}=C_{n-1}+v_{0}$ be a wheel of order $n$ with $V(W_{n})=\\{v_{i}|\
0\leq i\leq n-1\\}$ with $|V(W_{n})|=n$.

Case 1: When $n$ is even, assume the detour monophonic cover set
$D^{\prime}_{vc}=\\{v_{0},v_{i+j}|i=1,2,3,\cdots,n-1;j=0,1,2,\cdots,n-2\\}$,
where all the indices run over modulus $n$ and
$|D^{\prime}_{vc}|={\frac{n}{2}}+1$. Without loss of generality consider
${\dot{s}}=v_{1}$. Let the initial configuration be
${\sum_{i=\frac{n}{2}}^{n-3}2^{i}}+4$ pebbles assigned to cover the vertices
of $W_{n}$. Since the detour monophonic distance between $v_{0}$ and the
source vertex is 1, we can cover 2 pebbles. The vertices $v_{3}$, $v_{5}$ and
$v_{7}$ are covered with $2^{n-3}$, $2^{n-5}$ and $2^{n-7}$ pebbles
respectively. Proceeding in this manner we cover the all vertices of
$D^{\prime}_{vc}$ except the source vertex with
${\sum_{i=\frac{n}{2}}^{n-3}2^{i}}+4$ pebbles. Thus, continuing in this manner
we are left with $v_{1}$ uncovered. Hence,
$\mu_{vc}(W_{n})\geq{\sum_{i=\frac{n}{2}}^{n-3}2^{i}}+5$.
Now, consider $n$ is even and $n\equiv 0(mod~{}4)$.
Let the vertex cover set
$V(D^{1}_{vc})=\\{v_{0},v_{1},v_{3},v_{5},\cdots,v_{{\frac{n}{2}-3}},v_{\frac{n}{2}-1},v_{{\frac{n}{2}+1}},v_{{\frac{n}{2}+3}},\cdots,v_{n-3},\\\
v_{n-1}\\}$, where $|D^{1}_{vc}|={\frac{n}{2}}+1$. Fix ${\dot{s}}=v_{1}$.

| $v_{0}$ | $v_{1}$ | $v_{3}$ | $v_{5}$ | $\cdots$ | $v_{{\frac{n}{2}-3}}$ | $v_{\frac{n}{2}-1}$ | $v_{{\frac{n}{2}+1}}$ | $v_{{\frac{n}{2}+3}}$ | $\cdots$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ | $1$ | $0$ | $n-3$ | $n-5$ | $\cdots$ | $\frac{n}{2}+3$ | ${\frac{n}{2}}+1$ | $\frac{n}{2}$ | ${\frac{n}{2}}+2$ | $\cdots$ | $n-4$ | $1$
Table 20. Detour monophonic distance from $v_{1}\ to\ V(D^{1}_{vc})$.

Let $n$ is even and $n\equiv 2(mod~{}4)$.
Then the vertex cover set
$V(D^{1*}_{vc})=\\{v_{0},v_{1},v_{3},v_{5},\cdots,v_{{\frac{n}{2}-2}},v_{\frac{n}{2}},v_{{\frac{n}{2}+2}},v_{{\frac{n}{2}+4}},\cdots,v_{n-3},\\\
v_{n-1}\\}$ and $|D^{1*}_{vc}|={\frac{n}{2}}+1$. Without loss of generality
take ${\dot{s}}=v_{1}$.

| $v_{0}$ | $v_{1}$ | $v_{3}$ | $v_{5}$ | $\cdots$ | $v_{{\frac{n}{2}-2}}$ | $v_{\frac{n}{2}}$ | $v_{{\frac{n}{2}+2}}$ | $v_{{\frac{n}{2}+4}}$ | $\cdots$ | $v_{n-3}$ | $v_{n-1}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$v_{1}$ | $1$ | $0$ | $n-3$ | $n-5$ | $\cdots$ | ${\frac{n}{2}}+2$ | $\frac{n}{2}$ | ${\frac{n}{2}}+1$ | ${\frac{n}{2}}+3$ | $\cdots$ | $n-4$ | $1$
Table 21. Detour monophonic distance from $v_{1}\ to\ V(D^{1*}_{vc})$.

From Tables 20 and 21, in order to cover the vertices of $W_{n}$, it suffices
to cover the vertices of the corresponding vertex cover set. Thus, the pebble
distribution with either of the above vertex cover sets is given by,
$2^{1}+2^{0}+2^{{\frac{n}{2}}}+2^{{\frac{n}{2}}+1}+2^{{\frac{n}{2}}+2}+2^{{\frac{n}{2}}+3}+\cdots+2^{n-5}+2^{n-4}+2^{n-3}$,
which sums up into ${\sum_{i=\frac{n}{2}}^{n-3}2^{i}}+5$.
The result is true for all $v_{i}(2\leq i\leq n-1)$.
Let the source vertex be the hub. Then ${\dot{s}=v_{0}}$. Without loss of
generality let
$D^{4}_{vc}=\\{v_{0},v_{1},v_{3},\cdots,v_{n-5},v_{n-3},v_{n-1}\\}$.

The detour monophonic distance from any other vertex of $W_{n}$ to $v_{0}$ is
1. Thus in order to cover the vertices of $D^{4}_{vc}$, we have the pebble
distribution that sums up as follows.
$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{1})+\mu_{vc}(G,v_{3})+\cdots+\mu_{vc}(G,v_{n-5})+\mu_{vc}(G,v_{n-3})+\mu_{vc}(G,v_{n-1})$
$=2^{0}+2^{1}+2^{1}+\cdots+2^{1}+2^{1}+2^{1}$
$2{({\frac{n}{2}})}+1=n+1$ which is less than the considered number of
pebbles.

Thus, $\mu_{vc}(W_{n})={\sum_{i=\frac{n}{2}}^{n-3}2^{i}}+5$. Hence proved.

Case 2: When $W_{n}$ be an odd wheel of order $n$.
Let $(n\geq 3)$ and $n\equiv 1(mod~{}4)$.
Consider the detour monophonic vertex cover set
$D^{\prime\prime}_{vc}=\\{v_{0},v_{2(k)-1}|k=1,2,3,\\\ \cdots,n-2\\}$ and thus
$|D^{\prime\prime}_{vc}|={\lfloor\frac{n}{2}\rfloor}$.

Let ${\dot{s}}=v_{1}$ and the initial configuration be
$2\big{(}\sum_{j=\lfloor{\frac{n}{4}\rfloor}+1}^{{\lfloor{\frac{n}{2}}\rfloor}-1}2^{2(j)}\big{)}+2^{\lfloor{\frac{n}{2}}\rfloor}+1$.
In order to cover the vertices $v_{(n-3)}$ we require $2^{n-3}$ pebbles and to
cover $v_{(n-5)}$ we require $2^{n-5}$ pebbles respectively using the detour
monophonic path. Similar process can be carried on to cover the remaining
vertices of $D^{1}_{vc}$ using the following series of pebble distribution
that sums up,
$2^{1}+2^{n-3}+2^{n-5}+\cdots+2^{\lfloor\frac{n}{2}\rfloor+2}+2^{\lfloor\frac{n}{2}\rfloor}+2^{{\lfloor\frac{n}{2}\rfloor}+2}+\cdots+2^{n-5}+2^{n-3}$
$=2+2(2^{n-3}+2^{n-5}+\cdots+2^{{{\lfloor\frac{n}{2}\rfloor}+2}})+2^{\lfloor\frac{n}{2}\rfloor}$
$=2\big{(}\sum_{i=\lfloor{\frac{n}{4}\rfloor}+1}^{{\lfloor{\frac{n}{2}}\rfloor}-1}2^{2(i)}\big{)}+2^{\lfloor{\frac{n}{2}}\rfloor}+2$.
Thus we can cover the vertices $v_{3}$ and $v_{5}$ with $2^{n-3}$ and
$2^{n-5}$ pebbles and we are left with no pebble to cover the source vertex
$v_{1}$. Therefore, the detour monophonic vertex cover number
$\mu_{vc}(W_{n})\geq
2\big{(}\sum_{j=\lfloor{\frac{n}{4}\rfloor}+1}^{{\lfloor{\frac{n}{2}}\rfloor}-1}2^{2(j)}\big{)}+2^{\lfloor{\frac{n}{2}}\rfloor}+3$.

Now let ${\dot{s}}=v_{i}(1\leq i\leq n-1)$ and
$D^{2}_{vc}=\\{v_{0},v_{i+2},v_{i+4},\cdots,v_{\lfloor{\frac{n}{2}\rfloor}+i-2},v_{\lfloor{\frac{n}{2}\rfloor}+i},v_{{\lfloor{\frac{n}{2}\rfloor}+i+2}},\\\
\cdots,v_{n+i-5},v_{n+i-3}\\}$, where the indices run over modulus $n$.

| $v_{0}$ | $v_{i}$ | $v_{i+2}$ | $v_{i+4}$ | $\cdots$ | $v_{\lfloor{\frac{n}{2}\rfloor}+i-2}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+i}$ | $v_{{\lfloor{\frac{n}{2}\rfloor}+i+2}}$ | $\cdots$ | $v_{n+i-5}$ | $v_{n+i-3}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $1$ | $0$ | $n-3$ | $n-5$ | $\cdots$ | $\lfloor{\frac{n}{2}\rfloor}+2$ | $\lfloor{\frac{n}{2}}\rfloor$ | $\lfloor{\frac{n}{2}\rfloor}+2$ | $\cdots$ | $n-5$ | $n-3$
Table 22. Detour monophonic distance from $v_{i}$ to $V(D^{2}_{vc})$.

Obviously $v_{i}$ is covered with one pebble. From Table 22,
$D_{m}(v_{0},v_{i})=1$ and so $v_{0}$ is covered with two pebbles. In order to
cover the vertices $v_{i+2}$ and $v_{n+i-3}$ we require $2^{n-3}$ pebbles each
and to cover $v_{i+4}$ and $v_{n+i-5}$ we require $2^{n-5}$ pebbles each.
Similarly to cover $v_{\lfloor{\frac{n}{2}\rfloor}+i-2}$ and
$v_{\lfloor{\frac{n}{2}\rfloor}+i-2}$, we require $2^{\frac{n}{2}+2}$ pebbles.
We need $2^{\lfloor{\frac{n}{2}}\rfloor}$ pebbles to cover the vertex
$v_{{\lfloor{\frac{n}{2}\rfloor}+i}}$. Thus in order to cover the vertices in
$D^{2}_{vc}$ we distribute the pebbles as follows.
$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{i})+\mu_{vc}(G,v_{i+2})+\mu_{vc}(G,v_{i+4})+\cdots+\mu_{vc}(G,v_{{\lfloor{\frac{n}{2}}\rfloor}+i-2})+\mu_{vc}(G,v_{{\lfloor{\frac{n}{2}}\rfloor}+i})+\mu_{vc}(G,v_{i+{\frac{n+4}{2}}+2})+\cdots+\mu_{vc}(G,v_{n+i-5})+\mu_{vc}(G,v_{n+i-3})$.
$=2^{1}+2^{0}+2^{n-3}+2^{n-5}+\cdots+2^{\lfloor{\frac{n}{2}}\rfloor+2}+2^{\lfloor{\frac{n}{2}}\rfloor}+2^{\lfloor{\frac{n}{2}}\rfloor+2}+\cdots+2^{n-5}+2^{n-3}$
$=2+1+2(2^{n-3}+2^{n-5}+\cdots+2^{\lfloor{\frac{n}{2}}\rfloor+2})+2^{\lfloor{\frac{n}{2}}\rfloor}$
$=2\big{(}\sum_{j=\lfloor{\frac{n}{4}\rfloor}+1}^{{\lfloor{\frac{n}{2}}\rfloor}-1}2^{2(j)}\big{)}+2^{\lfloor{\frac{n}{2}}\rfloor}+3$.

Since all the vertices are of same detour monophonic distance, the number of
pebbles required to cover from any other vertex of $W_{n}$ except the hub is
$2\big{(}\sum_{j=\lfloor{\frac{n}{4}\rfloor}+1}^{{\lfloor{\frac{n}{2}}\rfloor}-1}2^{2(j)}\big{)}+2^{\lfloor{\frac{n}{2}}\rfloor}+3$.

Assume the source vertex is the hub.

Then ${\dot{s}=v_{0}}$. Without loss of generality let
$D*_{vc}=\\{v_{0},v_{1},v_{3},\cdots,v_{n-6},v_{n-4},v_{n-2}\\}$.

Since $D_{m}(x,v_{0})=1$, where $x$ is any vertex of $W_{n}$, Thus in order to
cover the vertices of $D*_{vc}$, we have the pebble distribution that sums up
as follows.
$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{1})+\mu_{vc}(G,v_{3})+\cdots+\mu_{vc}(G,v_{n-6})+\mu_{vc}(G,v_{n-4})+\mu_{vc}(G,v_{n-2})$
$=2^{0}+2^{1}+2^{1}+\cdots+2^{1}+2^{1}+2^{1}$
$=2{\lfloor{\frac{n}{2}}\rfloor}+1$ which is less than the considered number
of pebbles.
Thus,
$\mu_{vc}(W_{n})=2\big{(}\sum_{j=\lfloor{\frac{n}{4}\rfloor}+1}^{{\lfloor{\frac{n}{2}}\rfloor}-1}2^{2(j)}\big{)}+2^{\lfloor{\frac{n}{2}}\rfloor}+3$.
Hence proved.

Case 3:

When $n$ is odd and $n\equiv 3(mod~{}4)$, let
$D^{3}_{vc}=\\{v_{i+2j}|i=1,2,3,\cdots,n;j=0,1,2,\cdots,\lfloor{\frac{n}{4}-1}\rfloor,\lfloor{\frac{n}{4}}\rfloor,\lfloor{\frac{n}{4}+1}\rfloor,\cdots,{\frac{n}{2}-2},{\frac{n}{2}-1}\\}$
be the vertex cover set, where the indices run over modulus $n$. Thus,
$|D^{3}_{vc}|=\frac{n}{2}$. Now consider the configuration which encages
$2\big{(}\sum_{k}2^{2(k)}\big{)}+2$ pebbles to cover the vertices of $W_{n}$.
To cover the vertices $v_{i+2}$ and $v_{i+n-2}$, we need $2^{n-2}$ pebbles.
Similarly to cover the vertices $v_{i+4}$ and $v_{i+n-4}$, we need $2^{n-4}$
pebbles. We use $2^{\frac{n}{2}+1}$ pebbles to cover
$v_{i+2(\lfloor{\frac{n}{4}}\rfloor)}$ and
$v_{i+{2(\lfloor{\frac{n}{4}}\rfloor+1)}}$. Obviously, we require a pebble to
cover ${\dot{s}}=v_{i}(i=1,2,3,\cdots,n-1)$.

Thus, $\mu_{vc}(W_{n})\geq
2\big{(}\sum_{k={\lfloor{\frac{n}{4}}\rfloor}+1}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(k)}\big{)}+3$.

| $v_{i}$ | $v_{i+2}$ | $v_{i+4}$ | $\cdots$ | $v_{\lfloor{\frac{n}{2}\rfloor}+i-3}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+i-1}$ | $v_{\lfloor{\frac{n}{2}\rfloor}+i+1}$ | $v_{{\lfloor{\frac{n}{2}\rfloor}+i+3}}$ | $\cdots$ | $v_{n+i-5}$ | $v_{n+i-3}$
---|---|---|---|---|---|---|---|---|---|---|---
$v_{i}$ | $0$ | $n-3$ | $n-5$ | $\cdots$ | $\lfloor{\frac{n}{2}\rfloor}+3$ | $\lfloor{\frac{n}{2}\rfloor}+1$ | $\lfloor{\frac{n}{2}}\rfloor+1$ | $\lfloor{\frac{n}{2}\rfloor}+3$ | $\cdots$ | $n-5$ | $n-3$
Table 23. Detour monophonic distance from $v_{i}\ to\ V(D^{3}_{vc})$.

Clearly only one pebble suffices to cover the vertex
$v_{i}(i=1,2,3,\cdots,n)$. From Table 23, all the vertices of $D^{2}_{vc}$ are
covered by the following series of pebble distribution.

$\mu_{vc}(G,v_{0})+\mu_{vc}(G,v_{i})+\mu_{vc}(G,v_{i+2})+\mu_{vc}(G,v_{i+4})+\cdots+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}+i-3})+\\\
\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}+i-1})+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}+i+1})+\mu_{vc}(G,v_{\lfloor{\frac{n}{2}\rfloor}+i+3})+\cdots+\mu_{vc}(G,v_{n+i-5})+\mu_{vc}(G,v_{n+i-3})$.
$=2^{1}+2^{0}+2^{n-3}+2^{n-5}+\cdots+2^{\lfloor{\frac{n}{2}}\rfloor+3}+2^{\lfloor{\frac{n}{2}}\rfloor+1}+2^{\lfloor{\frac{n}{2}}\rfloor+1}+2^{\lfloor{\frac{n}{2}}\rfloor+3}+\cdots+2^{n-5}+2^{n-3}$
=$2\big{(}\sum_{k={\lfloor{\frac{n}{4}}\rfloor}+1}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(k)}\big{)}+3$.
Since all the vertices except $v_{0}$ of $W_{n}$ are at equivi distance, the
result holds for any $v_{i}\in V(W_{n})(2\leq i\leq n-1)$, when $n$ is odd and
$n\equiv 3(mod~{}4)$.

When the source vertex is the hub, the result follows from the previous
discussion of this Theorem and the configuration of pebbles is
$2{\lfloor{\frac{n}{2}}\rfloor}+1$ which is less than the considered number of
pebbles.
Hence,
$\mu_{vc}(W_{n})=2\big{(}\sum_{k={\lfloor{\frac{n}{4}}\rfloor}+1}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(k)}\big{)}+3$.

∎

##  3\. Algorithm to find DMVCPN

Step 1: Find the vertex cover set $D_{vc}$ with minimum cardinality for a
graph $G$, which covers all the edges of $G$.

Step 2: Fix the source vertex $\dot{s}$.

Step 3: Find the detour monophonic distance $D_{m}(\dot{s},v)$, where $v\in
V(G)$.

Step 4: By distributing the pebbles place one pebble each to cover the
vertices of $D_{vc}$ such that the number of pebbles used is minimum for all
configuration of pebbles.

Step 5: Repeat Step 4 for various source vertices until reaching the minimal
number of pebbles that cover all the vertices of $D_{vc}$ and so the vertices
of $G$.

Step 6: End the process if each vertex of $G$ is considered to be $\dot{s}$.

##  4\. Various Pebbling Numbers of $G$

The various graph pebbling numbers displayed in the Table 24 exhibit that
DMVCPN of a graph is distinct.

Standard | Detour pebbling | Monophonic pebbling | Detour monophonic vertex cover
---|---|---|---
Graph $G$ |  number $f^{*}(G)$ [7] |  number $\mu(G)$ [6] |  pebbling number $\mu_{vc}(G)$
$P_{n}$ | $2^{n-1}$ | $2^{n-1}$ |  $\sum_{i=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(i)+1},$ n is odd
|  |  |  $\sum_{j=0}^{{\frac{n}{2}}-1}2^{2(j)}$, n is even
$C_{n}$ | $2^{n-1}$ | $2^{n-2}+1$ | $\begin{cases}2\big{(}\sum_{i=\frac{n}{4}+1}^{\frac{n}{2}-1}2^{2(i)}\big{)}+2^{\frac{n}{2}}+1,&\text{if }n\ is\ \text{even }\text{and }n\equiv 0(mod~{}4)\\\ 2\big{(}\sum_{j={\lfloor\frac{n}{4}\rfloor}+1}^{\frac{n}{2}-1}2^{2(j)}\big{)}+1,&\text{if }n\ is\ \text{even }\text{and }\ n\equiv 2(mod~{}4)\\\ {\sum_{k={\lfloor\frac{n}{2}\rfloor}+1}^{n-2}2^{k}}+3,&\text{if }n\ is\ \text{odd }\par\end{cases}$
$F_{n}$ | $2^{n-1}$ | $2^{n-2}+1$ |  $\sum_{j=0}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(j)}+2$, n is odd
|  |  |  $\sum_{i=0}^{{\frac{n}{2}}-2}2^{2(i)+1}+2$, n is even
$W_{n}$ | $2^{n}$ | $2^{n-2}+2$ | $\begin{cases}{\sum_{i=\frac{n}{2}}^{n-3}2^{i}}+5,&\text{if }n\ is\ \text{even }\\\ 2\big{(}\sum_{j=\lfloor{\frac{n}{4}\rfloor}+1}^{{\lfloor{\frac{n}{2}}\rfloor}-1}2^{2(j)}\big{)}+2^{\lfloor{\frac{n}{2}}\rfloor}+3,&\text{if }n\ is\ \text{odd }\ \text{and }n\equiv 1(mod~{}4)\\\ 2\big{(}\sum_{k={\lfloor{\frac{n}{4}}\rfloor}+1}^{\lfloor{\frac{n}{2}}\rfloor-1}2^{2(k)}\big{)}+3,&\text{if }n\ is\ \text{odd}\ \text{and }\ n\equiv 3(mod~{}4)\\\ \end{cases}$
Table 24.  Various pebbling numbers for some standard graphs

##  5\. Conclusion

In this article, we have combined cover pebbling number and detour monophonic
number to obtain an interesting variate of graph theory, the detour monophonic
vertex cover pebbling number the interesting areas of graph theory. We have
studied and determined $\mu_{vc}(G)$ for some standard graphs. To evaluate the
specific values of $\mu_{vc}(G)$ for other classes of well-known graphs is an
open area of research.

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