The Convex Set Forming Game

Abstract

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., misère) variant, the last player to be able to select a vertex wins (resp., loses). We also define a new graph invariant $\mathrm{gc}\left(G\right)$ 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 misère), as well as the problem of deciding whether $\mathrm{gc}\left(G\right)\ge k$, are PSPACE-complete. As a by-product, we prove that the optimization variant of the classical Kayles game is PSPACE-complete. Then, we focus on convexable graphs, i.e., n-node graphs G for which $\mathrm{gc}\left(G\right)=n$. For this purpose, we say that a set $S=\{v_1,\cdots ,v_{\left|S\right|}\}\subseteq V$ in a graph G admits a Convex Elimination Ordering (CEO) if $\{v_1,\cdots ,v_i\}$ is convex for every $1\le i\le \left|S\right|$. 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 $\mathrm{gc}\left(G\right)$ in outerplanar graphs G.