Fractional Maker-Breaker Resolving Game

Abstract

Let G be a graph with vertex set V(G), and let d(u, w) denote the length of a $u-w$ geodesic in G. For two distinct $x,y \in V(G)$, let $R\{x,y\}=\{z \in V(G): d(x,z) \ne d(y,z)\}$. For a function g defined on V(G) and for $U \subseteq V(G)$, let $g(U)=\sum _{s\in U}g(s)$. A real-valued function $g: V(G) \rightarrow [0,1]$ is a resolving function of G if $g(R\{x,y\}) \ge 1$ for any distinct $x,y \in V(G)$. In this paper, we introduce the fractional Maker-Breaker resolving game (FMBRG). The game is played on a graph G by Resolver and Spoiler (denoted by $R^*$ and $S^*$, respectively) who alternately assigns non-negative real values on V(G) such that its sum is at most one on each turn. Moreover, the total value assigned, by $R^*$ and $S^*$, on each vertex over time cannot exceed one. $R^*$ wins if the total values assigned on V(G) by $R^*$, after finitely many turns, form a resolving function of G, whereas $S^*$ wins if $R^*$ fails to assign values on V(G) to form a resolving function of G. We obtain some general results on the outcome of the FMBRG and determine the outcome of the FMBRG for some graph classes.