[Paper Review] Cops and Robber Game with a Fast Robber on Interval, Chordal, and Planar Graphs

Abbas Mehrabian|arXiv (Cornell University)|Aug 25, 2010

Advanced Graph Theory Research13 references3 citations

TL;DR

This paper studies a variant of the Cops and Robber game where the robber moves with unbounded speed but cannot pass through vertices occupied by cops. It establishes tight bounds on the cop number $c_{ ext{∞}}(G)$ for interval, chordal, planar, and hypercube graphs, showing $c_{ ext{∞}}(G) = O(\sqrt{n})$ for interval graphs, $\Omega(n/\log n)$ for some chordal graphs, and $\Theta(\text{tw}(G))$ for planar graphs, with a 3-approximation algorithm for interval graphs and tight bounds for hypercubes.

ABSTRACT

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.

Motivation & Objective

• To analyze the cop number $c_{\infty}(G)$ in a variant of the Cops and Robber game where the robber has unbounded speed but cannot pass through occupied vertices.
• To determine tight upper and lower bounds for $c_{\infty}(G)$ in interval, chordal, planar, and hypercube graphs.
• To develop a polynomial-time 3-approximation algorithm for computing $c_{\infty}(G)$ in interval graphs.
• To explore the relationship between $c_{\infty}(G)$ and graph parameters like treewidth and maximum degree.
• To resolve open questions on the complexity and exact values of $c_{\infty}(G)$ for specific graph classes.

Proposed method

• Prove that for interval graphs, $c_{\infty}(G) = O(\sqrt{n})$ using structural decomposition and path-based strategies.
• Construct a polynomial-time 3-approximation algorithm for $c_{\infty}(G)$ in interval graphs by leveraging interval graph properties and dynamic programming.
• Establish a lower bound $c_{\infty}(G) \geq \frac{tw(G)+1}{\Delta(G)+1}$ using treewidth and maximum degree duality.
• Prove that for planar graphs (and graphs excluding a fixed apex minor), $c_{\infty}(G) = \Theta(tw(G))$, linking cop number to treewidth.
• Analyze the $m$-hypercube graph $\mathcal{H}_m$ using domination number and treewidth bounds to derive $c_{\infty}(\mathcal{H}_m) = \Theta(2^m / m)$.
• Use probabilistic methods and random graph theory to construct graphs with $c_{\infty}(G) = \Theta(n)$, showing linear cop number is possible.

Experimental results

Research questions

• RQ1 What is the exact asymptotic behavior of $c_{\infty}(G)$ for interval graphs, and can it be approximated efficiently?
• RQ2 Do there exist chordal graphs with $c_{\infty}(G) = \Omega(n)$, or is $\Omega(n/\log n)$ the tight upper bound?
• RQ3 Is $c_{\infty}(G) = \Theta(tw(G))$ for all graphs excluding a fixed apex minor, or only for planar graphs?
• RQ4 What is the exact value of $c_{\infty}(G)$ for the $m$-dimensional hypercube graph $\mathcal{H}_m$?
• RQ5 Is computing $c_{\infty}(G)$ in NP, and is the problem NP-hard for interval graphs?

Key findings

• For interval graphs, $c_{\infty}(G) = O(\sqrt{n})$, and this bound is tight, as shown by constructions achieving $\Omega(\sqrt{n})$.
• A polynomial-time 3-approximation algorithm exists for computing $c_{\infty}(G)$ in interval graphs.
• For every $n$, there exists a chordal graph $G$ with $c_{\infty}(G) = \Omega(n/\log n)$, showing the bound is nearly tight.
• For planar graphs (and graphs excluding a fixed apex minor), $c_{\infty}(G) = \Theta(tw(G))$, leading to an $O(1)$-approximation algorithm.
• For the $m$-hypercube graph $\mathcal{H}_m$, $c_{\infty}(\mathcal{H}_m) = \Theta(2^m / m)$, with bounds $\eta_1 2^m / (m\sqrt{m}) \leq c_{\infty}(\mathcal{H}_m) \leq \eta_2 2^m / m$ for constants $\eta_1, \eta_2 > 0$.
• The lower bound $c_{\infty}(G) \geq \frac{tw(G)+1}{\Delta(G)+1}$ holds for all graphs $G$, linking cop number to treewidth and maximum degree.