The competition-independence game with prevention

Abstract

The competition-independence game, as introduced by Phillips and Slater, is played on a graph by two players, Diminisher and Sweller, who are taking turns in choosing a vertex that is not in the closed neighborhood of any of the previously chosen vertices. The goal of Diminisher is to minimize the (maximal independent) set of chosen vertices at the end of the game, while Sweller wants just the opposite. Assuming that both players are playing optimally according to their goals, two graph invariants arise depending on who starts the game. In this paper, we introduce a variation of the game in which players are allowed at any stage in the game to use an alternative move called prevention. That is, a player can decide that in his/her move he/she will mark (not choose!) a previously unplayed vertex $x$ by which $x$ is prevented to be chosen during the rest of the game; in particular, $x$ is not in the final set of chosen vertices. Given a graph $G$, and assuming that both players play optimally accordingto their goals, ${\widetilde{I}_{\rm d}}(G)$ (resp.\ ${\widetilde{I}_{\rm s}}(G)$) denotes the size of the set of chosen vertices in the competition-independence game with prevention if Diminisher (resp.\ Sweller) moves first. By using the Partition Strategy we prove that for any positive integer $n$, ${\widetilde{I}_{\rm d}}(P_n)=\lfloor \frac{2n+3}{6}\rfloor$ and ${\widetilde{I}_{\rm s}}(P_n)=\lfloor \frac{2n+4}{6}\rfloor$. While it is not hard to establish the general bounds, $1\le {\widetilde{I}_{\rm d}}(G)\leq \lfloor \frac{n}{2}\rfloor$ and $1\le {\widetilde{I}_{\rm s}}(G)\leq \lceil \frac{n}{2}\rceil$, we characterize the classes of (connected) graphs $G$ that attain each of the four bounds. Finally, a close connection of the new game with a version of the coloring game called the packing coloring game is established for graphs with diameter $2$, and several open problems are posed.