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The damage number of the Cartesian product of graphs

Melissa A. Huggan, Margaret-Ellen Messinger, Amanda Porter

TL;DR

The paper investigates the damage number ${\rm dmg}(G)$ in a Cops and Robber variant on Cartesian products. It first proves a general upper bound for ${\rm dmg}(G \square H)$ and then derives exact results for products of finite trees, where ${\rm dmg}(T \square T') = {\rm rad}(T \square T') - 1$, and for products of cycles, where parity governs exactness (e.g., if at least one of $m,n$ is odd, ${\rm dmg}(C_m \square C_n) = \max\{ {\rm dmg}(C_m)|V(C_n)|, {\rm dmg}(C_n)|V(C_m)| \}$). It also characterizes graphs with small damage numbers (0, 1, 2) and analyzes how these values behave under Cartesian products, including a case where ${\rm dmg}(C_6 \square C_6) = 2|V(C_6)|+1$, which demonstrates that simple bounds are not always tight. The work introduces techniques such as relative capture times and shadow strategies to obtain tight bounds and exact values, contributing to pursuit-evasion theory and the understanding of product-graph structure on damage dynamics.

Abstract

We consider a variation of Cops and Robber, introduced in [D. Cox and A. Sanaei, The damage number of a graph, [Aust. J. of Comb. 75(1) (2019) 1-16] where vertices visited by a robber are considered damaged and a single cop aims to minimize the number of distinct vertices damaged by a robber. Motivated by the interesting relationships that often emerge between input graphs and their Cartesian product, we study the damage number of the Cartesian product of graphs. We provide a general upper bound and consider the damage number of the product of two trees or cycles. We also consider graphs with small damage number.

The damage number of the Cartesian product of graphs

TL;DR

The paper investigates the damage number in a Cops and Robber variant on Cartesian products. It first proves a general upper bound for and then derives exact results for products of finite trees, where , and for products of cycles, where parity governs exactness (e.g., if at least one of is odd, ). It also characterizes graphs with small damage numbers (0, 1, 2) and analyzes how these values behave under Cartesian products, including a case where , which demonstrates that simple bounds are not always tight. The work introduces techniques such as relative capture times and shadow strategies to obtain tight bounds and exact values, contributing to pursuit-evasion theory and the understanding of product-graph structure on damage dynamics.

Abstract

We consider a variation of Cops and Robber, introduced in [D. Cox and A. Sanaei, The damage number of a graph, [Aust. J. of Comb. 75(1) (2019) 1-16] where vertices visited by a robber are considered damaged and a single cop aims to minimize the number of distinct vertices damaged by a robber. Motivated by the interesting relationships that often emerge between input graphs and their Cartesian product, we study the damage number of the Cartesian product of graphs. We provide a general upper bound and consider the damage number of the product of two trees or cycles. We also consider graphs with small damage number.
Paper Structure (7 sections, 19 theorems, 22 equations, 3 figures)

This paper contains 7 sections, 19 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The damage number of products
  3. Products of general graphs
  4. Products of trees
  5. Products of cycles
  6. Graphs with small damage number
  7. Acknowledgements

Key Result

Theorem 1.4

For any graph $G$, ${\rm dmg}(G) \geq {\rm rad}(G)-1$.

Figures (3)

  • Figure 1: An example of initial positions on $C_6$.
  • Figure 2: An example of the set $D(C_{10})$ when $p=0$ and $x=4$ (rectangular region in left graph); $D'(C_{13})$ when $q=0$ and $y=7$ (grey region in top graph); and the corresponding sets $S_{C_{10}}$ (rectangular region in $C_{10} \square C_{13}$) and $S'_{C_{13}}$ (grey region in $C_{10} \square C_{13}$).
  • Figure 3: An example of a graph with radius $2$ and damage number $2$ where the optimal starting location for the cop is not a center vertex.

Theorems & Definitions (39)

  • Definition 1.1: CN2005
  • Definition 1.2: Dahl1995
  • Theorem 1.4: throttling
  • Proposition 2.1
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • ...and 29 more