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

TL;DR

This paper introduces the detour monophonic vertex cover pebbling number (DMVCPN), a new graph invariant, and computes it for standard graphs like cycles, paths, fans, and wheels, expanding pebbling theory.

Contribution

It defines DMVCPN, a novel concept in graph pebbling, and provides exact values for several fundamental graph classes, advancing understanding of pebbling dynamics.

Findings

DMVCPN computed for cycle, path, fan, and wheel graphs.

Establishes relationships between DMVCPN and graph structure.

Provides formulas and bounds for DMVCPN in standard graphs.

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 at 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 at least 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 required 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 the cycle, path, fan, and wheel graphs.