Restricted SDC Edge Cover Pebbling Number

TL;DR

This paper introduces and computes the restricted SDC edge cover pebbling number, a new graph invariant involving pebbling distributions with parity restrictions on edges, expanding understanding of pebbling dynamics under constraints.

Contribution

It defines the restricted SDC edge cover pebbling number and calculates this invariant for specific classes of graphs, providing new insights into constrained pebbling problems.

Findings

Computed the restricted SDC edge cover pebbling number for certain graphs.

Established bounds and exact values for specific graph classes.

Extended pebbling theory to include parity-restricted distributions.

Abstract

The restricted edge pebbling distribution is a distribution of pebbles on the edges of $G$ is the placement of pebbles on the edges with the restriction that only an even number of pebbles should be placed on the edges with labels $0$. Given an SDC labeling of $G$, the restricted SDC edge cover pebbling number of a graph $G$, $\psi_{EC}(G)$, is the least positive integer $m$ for which any restricted edge pebbling distribution of $m$ pebbles such that at the end there are no pebbles on the edges having label $0$ will allow the shifting of a pebble simultaneously to all edges with label $1$ using a sequence of restricted edge pebbling moves. We compute the restricted SDC edge cover pebbling number for some graphs