Domination Cover Pebbling: Graph Families

Abstract

Given a configuration of pebbles on the vertices of a connected graph \( G \), a \emph{pebbling move} is defined as the removal of two pebbles from some vertex and the placement of one of these on an adjacent vertex. We introduce the notion of domination cover pebbling, obtained by combining graph cover pebbling with the theory of domination in graphs. The domination cover pebbling number, \( \psi(G) \), of a graph \( G \) is the minimum number of pebbles that are placed on \( V(G) \) such that after a sequence of pebbling moves, the set of vertices with pebbles forms a dominating set of \( G \), regardless of the initial configuration of pebbles. We discuss basic results and determine \( \psi(G) \) for paths, cycles, and complete binary trees.