## Andres, Dominique and Lock, Edwin - Characterising and recognising game-perfect graphs

dmtcs:4935 - Discrete Mathematics & Theoretical Computer Science, May 23, 2019, vol. 21 no. 1, ICGT 2018
Characterising and recognising game-perfect graphs

Authors: Andres, Dominique and Lock, Edwin

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.

Volume: vol. 21 no. 1, ICGT 2018
Published on: May 23, 2019
Submitted on: October 31, 2018
Keywords: Mathematics - Combinatorics,05C57