The Cops & Robber game on series-parallel graphs

TL;DR

This paper proves that series-parallel graphs are 2-copwin in the Cops and Robber game, extending known results for planar and outerplanar graphs by introducing a variant with exits and providing a winning strategy.

Contribution

It establishes that all series-parallel graphs are 2-copwin, a new result extending the class of graphs with known cop-win properties, using a novel game variant and strategy.

Findings

Series-parallel graphs are 2-copwin.

A new variant of the game with exits is introduced.

The proof provides an explicit winning strategy for the cops.

Abstract

The Cops and Robber game is played on undirected finite graphs. $k$ cops and one robber are positioned on vertices and take turn in moving along edges. The cops win if, after a move, a cop and the robber are on the same vertex. A graph is called $k$-copwin, if the cops have a winning strategy. It is known that planar graphs are 3-copwin (Aigner & Fromme, 1984) and that outerplanar graphs are 2-copwin (Clarke, 2002). In this short note, we prove that series-parallel (i.e., graphs with no $K_4$ minor) graphs are 2-copwin. It is a well-known trick in the literature of cops & robber games to define variants of the game which impose restrictions on the possible strategies of the cops (see Clarke, 2002). For our proof, we define the ``cops & robber game with exits''. Our proof yields a winning strategy for the cops.