[Paper Review] The Cops and Robber game on graphs with forbidden (induced) subgraphs

Gwenaël Joret, Marcin Kamiński+1 more|arXiv (Cornell University)|Apr 25, 2008

Advanced Graph Theory Research14 references33 citations

TL;DR

This paper characterizes the cop number in graph classes defined by forbidding a single graph as a subgraph or induced subgraph. It proves that the cop number is bounded if and only if all connected components of the forbidden graph are paths (for induced subgraphs) or trees with at most three leaves (for subgraphs), using strategies based on distance reduction and tree-decomposition techniques. The key contribution is a complete structural characterization of graphs with bounded cop number under forbidden subgraph constraints.

ABSTRACT

The two-player, complete information game of Cops and Robber is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. In this paper, we study the cop number in the classes of graphs defined by forbidding one or more graphs as either subgraphs or induced subgraphs. In the case of a single forbidden graph we completely characterize (for both relations) the graphs which force bounded cop number. In closing, we bound the cop number in terms of the tree-width.

Motivation & Objective

• To determine the structural conditions under which the cop number remains bounded in graph classes defined by forbidding a single graph as a subgraph or induced subgraph.
• To generalize known results on planar and minor-free graphs by identifying the exact forbidden subgraphs that force bounded cop number.
• To establish tight connections between cop number, tree-width, and graph circumference, providing new algorithmic and structural insights.
• To resolve open problems on combinatorial characterizations of graphs with bounded cop number by identifying forbidden subgraph obstructions.

Proposed method

• Uses a staged strategy where cops reduce the distance to the robber by moving along shortest paths and maintaining a path-like formation.
• Applies tree-decomposition techniques to simulate a cop strategy that guards cutsets between adjacent tree-bags, ensuring the robber cannot escape.
• Employs inductive reasoning on the number of connected components of the forbidden graph, reducing the problem to base cases on trees with bounded degree.
• Leverages known bounds on cop number in terms of tree-width (cop(G) ≤ tw(G)/2 + 1) and relates this to graph circumference via tw(G) ≤ circ(G) − 1.
• Uses the concept of 'bibj-paths' and guarded paths to maintain control over critical vertices during cop movements.
• Applies a subdivision argument to construct graphs with unbounded cop number when forbidden graphs contain high-degree vertices or multiple degree-3 nodes.

Experimental results

Research questions

• RQ1 For which graphs H is the cop number bounded in the class of H-free (induced) graphs?
• RQ2 What structural properties of H ensure that all H-free graphs have bounded cop number?
• RQ3 How does the cop number relate to tree-width and graph circumference in H-free graphs?
• RQ4 Can the cop number be unbounded even when H is a forest, and if so, under what conditions?
• RQ5 What is the precise characterization of graphs H such that H-subgraph-free graphs have bounded cop number?

Key findings

• The class of H-free graphs has bounded cop number if and only if every connected component of H is a path.
• The class of H-subgraph-free graphs has bounded cop number if and only if every connected component of H is a tree with at most three leaves.
• For any ℓ ≥ 3, every Pℓ-free graph has cop number at most ℓ − 2, and this bound is tight for certain graph families.
• The cop number of a graph is at most half its tree-width plus one, and this bound is sharp for tree-width up to 5.
• The cop number is at most half the circumference of the graph, a consequence of the tree-width and circumference relationship.
• Graphs with unbounded cop number exist even under strong constraints (e.g., 3-regular graphs), showing that maximum degree 3 is not sufficient to bound cop number unless structural constraints on component trees are met.