Cops and robbers pebbling in graphs

TL;DR

This paper introduces Cops and Robbers Pebbling, a new graph theory paradigm combining elements of Cops and Robbers and Graph Pebbling, analyzing cop pebbling numbers across various graph classes and exploring related conjectures.

Contribution

It defines the cop pebbling number, provides bounds and exact values for different graphs, and investigates conjectures related to cop pebbling and existing graph theory problems.

Findings

Upper and lower bounds for cop pebbling numbers on various graphs

Exact cop pebbling numbers for specific graph classes

Failure of Graham's Pebbling Conjecture inequality in cop pebbling

Abstract

Here we merge the two fields of Cops and Robbers and Graph Pebbling to introduce the new topic of Cops and Robbers Pebbling. Both paradigms can be described by moving tokens (the cops) along the edges of a graph to capture a special token (the robber). In Cops and Robbers, all tokens move freely, whereas, in Graph Pebbling, some of the chasing tokens disappear with movement while the robber is stationary. In Cops and Robbers Pebbling, some of the chasing tokens (cops) disappear with movement, while the robber moves freely. We define the cop pebbling number of a graph to be the minimum number of cops necessary to capture the robber in this context, and present upper and lower bounds and exact values, some involving various domination parameters, for an array of graph classes, including paths, cycles, trees, chordal graphs, high girth graphs, and cop-win graphs, as well as graph products. Furthermore we show that the analogous inequality for Graham's Pebbling Conjecture fails for cop pebbling and posit a conjecture along the lines of Meyniel's Cops and Robbers Conjecture that may hold for cop pebbling. We also offer several new problems.