P_3-Games on Chordal Bipartite Graphs

Let G = (V ,E) be a connected graph. A set U ⊆ V is convex if G[U] is connected and all vertices of V \U have at most one neighbor inU. Let σ(W) denote the unique smallest convex set that contains W ⊆ V . Two players play the following game. Consider a convex set U and call it the ‘playground.’ Initially, U = ∅. When U = V , the player to move loses the game. Otherwise, that player chooses a vertex x ∈ V \ U which is at distance at most two fromU. The effect of the move is that the playground U changes into σ(U ∪ {x}) and the opponent is presented with this new playground. A graph is chordal bipartite if it is bipartite and has no induced cycle of length more than four. In this paper we show that, when G is chordal bipartite, there is a polynomial-time algorithm that computes the Grundy number of the P3-game played on G. This implies that there is an efficient algorithm to decide whether the first player has a winning strategy.