Characterising and recognising game-perfect graphs

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 colouran uncoloured vertex such that adjacent vertices receive different colours. Thegame ends once the graph is fully coloured, in which case the maker wins, orthe graph can no longer be fully coloured, in which case the breaker wins. Inthe game $g_B$, the breaker makes the first move. Our main focus is on theclass 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, wecharacterise $g_B$-perfect graphs in two ways, by forbidden induced subgraphsand by explicit structural descriptions. We also present a clique moduledecomposition, which may be of independent interest, that allows us toefficiently recognise $g_B$-perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the International Colloquium on Graph Theory (ICGT) 201