Equality Perfect Graphs and Digraphs

In the graph colouring game introduced by Bodlaender [7], two players, Alice and Bob, alternately colour uncoloured vertices of a given graph $G$ with one of $k$ colours so that adjacent vertices receive different colours. Alice wins if every vertex is coloured at the end. The game chromatic number of $G$ is the smallest $k$ such that Alice has a winning strategy. In Bodlaender's original game, Alice begins. We also consider variants of this game where Bob begins or skipping turns is allowed [1] and their generalizations to digraphs [2]. By means of forbidden induced subgraphs (resp.\ forbidden induced subdigraphs), for several pairs $(g_1,g_2)$ of such graph (resp.\ digraph) colouring games $g_1$ and~$g_2$, which define game chromatic numbers $\chi_{g_1}$ and~$\chi_{g_2}$, we characterise the classes of graphs (resp.\ digraphs) such that, for any induced subgraph (resp.\ subdigraph)~$H$, the game chromatic numbers $\chi_{g_1}(H)$ and $\chi_{g_2}(H)$ of~$H$ are equal.