Cops and robbers pebbling in graphs

TL;DR

This work introduces Cops and Robbers Pebbling, defining the cop pebbling number $\pi^{\sc}(G)$ as the minimum number of cops needed to capture a robber when cops move by pebbling steps and the robber moves freely. It systematically derives lower and upper bounds, exact values, and relationships to domination, optimal pebbling, and graph products across a broad range of graph classes, including paths, cycles, trees, chordal and high-girth graphs, and Cartesian products. A key result is that the standard Graham’s Pebbling conjecture inequality fails in the cop-pebbling setting, motivating Meyniel-type conjectures and several open problems. The paper also provides detailed analyses for Cartesian products such as ladders and grids, and highlights both positive product bounds (e.g., $\pi^{\sc}(G\Box K_t) \le t\pi^{\sc}(G)$) and counterexamples (e.g., $G^d$ and wheels) that separate cop pebbling from classical pebbling. Overall, it lays a foundation for further exploration of cop pebbling through domination, product structures, and open questions about constant bounds and asymptotic behavior.

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.

Theorems & Definitions (54)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof