The Convex Set Forming Game
Résumé
In 1984, Frank Harary introduced the first graph convexity game, focused on the geodesic convexity. A set $S \subseteq V$ of vertices of a graph $G=(V,E)$ is convex if every shortest path between two vertices of $S$ is also included in $S$. We introduce the Convex Set Forming Game $\CFG$: two players alternately select vertices in such a way that the set of selected vertices is always a convex set. In the normal (resp., mis\`ere) variant, the last player to be able to select a vertex wins (resp., loses). We also define a new graph invariant $\gcon(G)$ as the largest integer $k$ such that the first player has a strategy ensuring that, at the end of the game, at least $k$ vertices of the graph $G$ have been selected. We first show that the problems of deciding the outcome (does the first player win?) of the game in both variants (normal and mis\`ere), as well as the problem of deciding whether $\gcon(G)\geq k$, are PSPACE-complete. As a by-product, we prove that the optimization variant of the classical \textsc{Kayles} game is PSPACE-complete. Then, we focus on convexable graphs, i.e., $n$-node graphs $G$ for which $\gcon(G)=n$. For this purpose, we say that a set $S=\{v_1,\cdots,v_{|S|}\}\subseteq V$ in a graph $G$ admits a Convex Elimination Ordering (CEO) if $\{v_1,\cdots,v_i\}$ is convex for every $1 \leq i \leq |S|$. We show that the class of graphs whose vertex-set admits a CEO coincides with the chordal graphs and that this class strictly contains the convexable graphs. Moreover, every graph which is Ptolemaic (distance-hereditary chordal) or unit interval is convexable. Finally, we give a polynomial-time algorithm for computing a largest set admitting a CEO in outerplanar graphs, which gives upper bounds on $\gcon(G)$ in outerplanar graphs $G$.
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