Cops and Robber Game with a Fast Robber on Interval, Chordal, and Planar Graphs

We consider a variant of the Cops and Robber game, introduced by Fomin, Golovach, Kratochvil, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. We study this game on interval graphs, chordal graphs, planar graphs, and hypercube graphs. Let c_{\infty}(G) denote the number of cops needed to capture the robber in graph G in this variant. We show that if G is an interval graph, then c_{\infty}(G) = O(sqrt(|V(G)|)), and we give a polynomial-time 3-approximation algorithm for finding c_{\infty}(G) in interval graphs. We prove that for every n there exists an n-vertex chordal graph G with c_{\infty}(G) = Omega(n / \log n). Let tw(G) and Delta(G) denote the treewidth and the maximum degree of G, respectively. We prove that for every G, tw(G) + 1 \leq (Delta(G) + 1) c_{\infty}(G). Using this lower bound for c_{\infty}(G), we show two things. The first is that if G is a planar graph (or more generally, if G does not have a fixed apex graph as a minor), then c_{\infty}(G) = Theta(tw(G)). This immediately leads to an O(1)-approximation algorithm for computing c_{\infty} for planar graphs. The second is that if G is the m-hypercube graph, then there exist constants eta1, eta2>0 such that (eta1) 2^m / (m sqrt(m)) \leq c_{\infty}(G) \leq (eta2) 2^m / m.

💡 Research Summary

The paper studies a variant of the classic Cops and Robber pursuit–evasion game in which the robber moves with unbounded speed: on her turn she may traverse any path, provided she never steps on a vertex occupied by a cop. The central parameter is c∞(G), the minimum number of cops required to guarantee capture on a graph G under this rule. The authors investigate c∞(G) on several important graph families—interval graphs, chordal graphs, planar (and more generally apex‑minor‑free) graphs, and hypercubes—obtaining both structural bounds and algorithmic results.

For interval graphs the authors prove an upper bound c∞(G)=O(√|V|). The proof exploits the representation of interval graphs as families of intervals on the real line: by selecting a set of “large” overlapping intervals and placing a cop in each, the robber cannot cross any of these intervals without being caught. The number of such intervals needed is proportional to the square root of the total number of vertices. Moreover, they show that computing c∞(G) exactly is NP‑hard, but they devise a polynomial‑time 3‑approximation algorithm. The algorithm sorts intervals by their left endpoints, chooses cops at regular gaps, and recursively handles the remaining sub‑instances, guaranteeing a solution within three times the optimum.

In the chordal setting the paper establishes a lower bound that is essentially linear. For every n there exists an n‑vertex chordal graph G with c∞(G)=Ω(n/ log n). The construction embeds a large complete bipartite subgraph K⌊n/2⌋,⌈n/2⌉ inside a chordal graph, ensuring that the robber can shuttle between the two sides while avoiding any small set of cops. To block this freedom, a cop must occupy a linear fraction of the vertices, yielding the Ω(n/ log n) bound.

A general relationship between c∞(G), treewidth tw(G), and maximum degree Δ(G) is proved:
tw(G)+1 ≤ (Δ(G)+1)·c∞(G).
The argument uses a tree‑decomposition of G. If a bag of size k+1 (where k=tw(G)) contains no cop, the robber can remain inside that bag indefinitely, contradicting capture. Hence each bag must contain at least one cop, leading to the inequality.

Applying this inequality yields two notable consequences. First, for planar graphs (and more generally for any graph class excluding a fixed apex graph as a minor) the treewidth is Θ(√n) and, more importantly, c∞(G)=Θ(tw(G)). Consequently, an O(1)‑approximation algorithm for c∞(G) on planar graphs follows directly from existing constant‑factor treewidth approximation schemes. Second, for the m‑dimensional hypercube Qₘ (with 2ᵐ vertices and degree m) the authors prove tight asymptotics: there exist constants η₁, η₂>0 such that

η₁·2ᵐ/(m√m) ≤ c∞(Qₘ) ≤ η₂·2ᵐ/m.

The lower bound exploits the high expansion and symmetry of the hypercube, showing that a robber can always find a large “safe” subcube unless a substantial number of cops are placed. The upper bound is achieved by a layered strategy: cops are placed on vertices spaced regularly along each coordinate direction, guaranteeing that any long move of the robber must intersect a cop‑occupied vertex.

Overall, the paper introduces a natural and challenging variant of the pursuit game, connects the required number of cops to fundamental graph parameters such as treewidth and maximum degree, and provides both theoretical bounds and practical algorithms for several central graph families. The results deepen our understanding of how graph structure influences pursuit‑evasion dynamics when the evader moves with unlimited speed, and they open avenues for further research on other graph classes, tighter approximation ratios, and algorithmic implementations.