Cop number of partial cubes
Nicholas Crawford, Vesna Iršič Chenoweth
TL;DR
The authors address the cop-number problem on partial cubes, introducing necessary concepts and focusing on Fibonacci and Lucas cubes. They prove a general lower bound $c(G) \geq \left\lceil \frac{\delta(G)}{2} \right\rceil$ via a distance-2 neighbor lemma and derive improved upper bounds for median graphs, along with tight-ish bounds for Fibonacci and Lucas cubes: $\left\lfloor \frac{n+5}{6} \right\rfloor \le c(\Gamma_n), c(\Lambda_n) \le \left\lceil \frac{n}{4} \right\rceil$, with further refinements $c(\Gamma_n) \le \left\lceil \frac{n}{3} \right\rceil$ for $n \ge 6$ and $c(\Gamma_n) \le \left\lceil \frac{n}{4} \right\rceil$ for $n \ge 9$. The work combines structural properties of isometric subgraphs with constructive strategies (block partitioning) to advance understanding of pursuit-evasion on these graphs. It also links these graph-theoretic results to applications in chemical graph theory via resonance graphs. The results establish a framework for bounding cop numbers on partial cubes and motivate future study of broader graph families within this class.
Abstract
The game of Cops and Robbers on graphs is a well-studied pursuit--evasion model whose central parameter, the cop number, captures the minimum number of pursuers required to guarantee capture of an adversary on a given graph. While the cop number has been determined for many classical graph families, relatively little is known about the important class of partial cubes, i.e., isometric subgraphs of hypercubes. In this paper, we establish a lower bound for the cop number of partial cubes and present an upper bound on a subclass of partial cubes. Additionally, we improve these bounds for a particular family of partial cubes: Fibonacci cubes. These graphs are defined as induced subgraphs of hypercubes obtained by forbidding consecutive ones in binary strings. Beyond their natural combinatorial interest, Fibonacci cubes have connections to chemical graph theory, where they serve as models for resonance graphs of certain classes of polycyclic aromatic hydrocarbons.