Characterising and recognising game-perfect graphs

TL;DR

This paper characterizes and recognizes a class of graphs called g_B-perfect graphs, which guarantee a winning strategy for the maker in a vertex colouring game played on any induced subgraph, using structural and forbidden subgraph characterizations.

Contribution

It provides the first complete structural and forbidden subgraph characterizations of g_B-perfect graphs, along with an efficient recognition algorithm based on clique module decomposition.

Findings

Characterization of g_B-perfect graphs via forbidden induced subgraphs.

Explicit structural description of g_B-perfect graphs.

Efficient recognition algorithm using clique module decomposition.

Abstract

Consider a vertex colouring game played on a simple graph with $k$ permissible colours. Two players, a maker and a breaker, take turns to colour an uncoloured vertex such that adjacent vertices receive different colours. The game ends once the graph is fully coloured, in which case the maker wins, or the graph can no longer be fully coloured, in which case the breaker wins. In the game $g_B$, the breaker makes the first move. Our main focus is on the class of $g_B$-perfect graphs: graphs such that for every induced subgraph $H$, the game $g_B$ played on $H$ admits a winning strategy for the maker with only $\omega(H)$ colours, where $\omega(H)$ denotes the clique number of $H$. Complementing analogous results for other variations of the game, we characterise $g_B$-perfect graphs in two ways, by forbidden induced subgraphs and by explicit structural descriptions. We also present a clique module decomposition, which may be of independent interest, that allows us to efficiently recognise $g_B$-perfect graphs.