[Paper Review] Surrounding cops and robbers on graphs of bounded genus

Peter Bradshaw, Seyyed Aliasghar Hosseini|arXiv (Cornell University)|Sep 22, 2019

Advanced Graph Theory Research16 references4 citations

TL;DR

This paper establishes tight bounds on the surrounding cop number for graphs of bounded genus, showing that planar graphs require at most 7 cops and toroidal graphs at most 8, with improved bounds for bipartite cases. The authors use a geodesic path guarding strategy with 2–3 cops per path to iteratively reduce the robber’s territory until surrounded.

ABSTRACT

We consider a surrounding variant of cops and robbers on graphs of bounded genus. We obtain bounds on the number of cops required to surround a robber on planar graphs, toroidal graphs, and outerplanar graphs. We also obtain improved bounds for bipartite planar and toroidal graphs. We briefly consider general graphs of bounded genus and graphs with a forbidden minor.

Motivation & Objective

• To determine the minimum number of cops required to surround a robber on graphs of bounded genus, particularly planar and toroidal graphs.
• To extend existing cop number strategies to the surrounding variant of cops and robbers, where victory is achieved by isolating the robber via surrounding cops.
• To improve bounds for bipartite planar and toroidal graphs, and to generalize results to graphs excluding a minor.
• To establish tight upper bounds and demonstrate their tightness via explicit constructions.

Proposed method

• Employ geodesic path guarding as the core strategy, where cops are positioned to prevent the robber from accessing specific paths.
• Use three cops to guard a geodesic path in general graphs, and two cops in bipartite graphs, based on distance shadowing and path control.
• Apply iterative reduction of the robber’s territory by successively guarding geodesic paths, shrinking the connected component containing the robber.
• Leverage known results on cop number for graphs of genus g and minor-free graphs, adapting them to the surrounding variant using guarded path strategies.
• Use distance-based shadowing: cops track the robber’s proximity to a path’s endpoints to prevent access.
• Prove tightness of bounds via explicit constructions, such as the 8-vertex grid $P_1 \square P_3$, showing $s(G) = 3$ for bipartite outerplanar graphs.

Experimental results

Research questions

• RQ1 What is the maximum surrounding cop number for planar graphs, and is 7 tight?
• RQ2 What is the maximum surrounding cop number for toroidal graphs, and is 8 tight?
• RQ3 Can improved bounds be established for bipartite planar and toroidal graphs using the same guarding strategy?
• RQ4 How do surrounding cop number bounds generalize to graphs of higher genus and those excluding a minor?
• RQ5 What is the impact of movement restrictions (e.g., limited moving cops per turn) on the surrounding cop number?

Key findings

• The surrounding cop number of any planar graph is at most 7, and this bound is tight as there exist planar graphs requiring 6 cops.
• For toroidal graphs, the surrounding cop number is at most 8, with examples showing it can be as high as 7.
• For bipartite planar and toroidal graphs, improved bounds are established: $s(G) \leq 4$ and $s(G) \leq 5$, respectively.
• For graphs of genus $g$, the surrounding cop number is bounded by $s(G) \leq 4g + 10$, and for bipartite graphs, $s(G) \leq \lfloor \frac{8}{3}g + \frac{20}{3} \rfloor$.
• For graphs excluding a minor $H$, the surrounding cop number is bounded by $s(G) \leq 3|E(H - h)|$ for general graphs and $s(G) \leq 2|E(H - h)|$ for bipartite graphs.
• A bipartite outerplanar graph $G = P_1 \square P_3$ is constructed with $s(G) = 3$, demonstrating the tightness of the bound for outerplanar graphs.