[Paper Review] The Cop Number of Graphs with Forbidden Induced Subgraphs

Mingrui Liu|arXiv (Cornell University)|Aug 29, 2019

Advanced Graph Theory Research5 references4 citations

TL;DR

This paper investigates the cop number of graphs that exclude certain induced subgraphs, introducing a novel method to bound the cop number for families of graphs forbidden from containing specific path-like or star-like structures. It proves that $c(G) \leq k-2$ for $P_k$-free graphs ($k \geq 3$), extends this to $(P_i + P_j)$-free graphs with $c(G) \leq i+j-2$, and establishes tighter bounds for $\{P_k, K_{1,3}\}$-free and $\{2P_2, \overline{P_5}\}$-free graphs, resolving a conjecture partially and generalizing prior results.

ABSTRACT

In the game of Cops and Robber, a team of cops attempts to capture a robber on a graph $G$. Initially, all cops occupy some vertices in $G$ and the robber occupies another vertex. In each round, a cop can move to one of its neighbors or stay idle, after which the robber does the same. The robber is caught by a cop if the cop lands on the same vertex which is currently occupied by the robber. The minimum number of cops needed to guarantee capture of a robber on $G$ is called the {\em cop number} of $G$, denoted by $c(G)$. We say a family $\cal F$ of graphs is {\em cop-bounded} if there is a constant $M$ so that $c(G)\leq M$ for every graph $G\in \cal F$. Joret, Kaminński, and Theis [Contrib. Discrete Math. 2010] proved that the class of all graphs not containing a graph $H$ as an induced subgraph is cop-bounded if and only if $H$ is a linear forest; morerover, $C(G)\leq k-2$ if if $G$ is induced-$P_k$-free for $k\geq 3$. In this paper, we consider the cop number of a family of graphs forbidding certain two graphs and generalized some previous results.

Motivation & Objective

• To determine the cop number for graphs that exclude specific induced subgraphs, particularly paths and claws.
• To generalize prior results on $P_k$-free graphs to families forbidding multiple induced subgraphs.
• To provide a unified method for bounding cop numbers using structural graph decomposition and pursuit strategies.
• To resolve a conjecture by Sivaraman and Testa regarding the cop number of $2P_2$-free graphs.

Proposed method

• A new strategy is developed based on controlling distances between cops and the robber using shortest path structures and neighborhood constraints.
• The proof technique leverages the concept of minimal distance $f$ between cops and robber, analyzing its evolution over rounds.
• Cops are strategically positioned on vertices and neighborhoods to isolate the robber in components that are $P_k$-free.
• The method uses induction and structural analysis to show that $k-2$ cops suffice for $P_k$-free graphs, extending to unions of paths.
• For $\{P_k, K_{1,3}\}$-free graphs, the approach combines path-freeness with claw-freeness to reduce the cop number to $k-3$.
• The method is generalized to $(P_{i_1} + \cdots + P_{i_k})$-free graphs, yielding a bound of $\sum i_j - 2$ cops.

Experimental results

Research questions

• RQ1 What is the maximum cop number for graphs that are $P_k$-free for $k \geq 3$?
• RQ2 Can the cop number be bounded for graphs that exclude a pair of induced subgraphs, such as $P_k$ and $K_{1,3}$?
• RQ3 How does the cop number behave in graphs that are $P_i + P_j$-free for $i,j \geq 1$?
• RQ4 Can the cop number be reduced when forbidding induced subgraphs that are disjoint unions of paths?
• RQ5 What is the cop number of $\{2P_2, \overline{P_5}\}$-free graphs, and how does it compare to known bounds?

Key findings

• The cop number of any $P_k$-free graph is at most $k-2$ for $k \geq 3$, confirming a known result with a new proof method.
• For $\{P_k, K_{1,3}\}$-free graphs with $k \geq 5$, the cop number is at most $k-3$, improving upon the $P_k$-free bound.
• The cop number of $(P_i + P_j)$-free graphs is at most $i + j - 2$, showing that forbidding disjoint unions of paths yields a tighter bound than forbidding individual paths.
• For $\{2P_2, \overline{P_5}\}$-free graphs, the cop number is at most 2, generalizing earlier results on $2P_2$-free graphs with triangle-free complements.
• The paper proves $c(G) \leq k-1$ for $(P_1 + P_k)$-free graphs and $c(G) \leq k$ for $(P_2 + P_k)$-free graphs, extending the applicability of the method.
• The method successfully resolves a partial case of a conjecture by Sivaraman and Testa on the cop number of $2P_2$-free graphs.