Fashion Game on Planar Graphs

Abstract

This paper studies an optimization problem of the fashion game on graphs on surfaces, especially on planar graphs. There are rebel players in a graph G. All players choose their actions from an identical set of the two symmetric actions, say $\{0,1\}$. An action profile of G is a mapping $\pi :V(G)\rightarrow \{0,1\}$. A rebel likes people having the different action with her and dislikes people having the same action. The utility $u(v,\pi )$ of a player v under the action profile $\pi$ is the number of neighbors she likes minus the number of neighbors she dislikes. Let $\phi :V(G)\rightarrow {\mathbb {Z}}$ be a function. The $\phi$-satisfiability problem is to determine whether a graph has an action profile under which each player v has a utility at least $\phi (v)$. Let t be an integer. The t-satisfiability problem is the specialized $\phi$-satisfiability problem when $\phi (v)=t$, for each $v\in V(G)$. The utility of G, denoted by u(G), is defined to be the maximum t such that G is t-satisfiable. Let $\eta : V(G)\rightarrow {\mathbb {N}}$ be a function. A mapping $c:\ V(G)\rightarrow \{0,1\}$ is called an $\eta$-defective 2-coloring of G if every $v\in V(G)$ has at most $\eta (v)$ neighbors that have the same color with it. For graphs embeddable in surfaces, upper bounds of their utilities are given. The graphs embeddable in the torus or the Klein bottle whose utilities reach their upper bounds are determined. The t-satisfiability problem for graphs embeddable in the plane, the projective plane, the torus, or the Klein bottle is NP-complete if $t\in \{1,2,3\}$, and is polynomial time solvable otherwise. We design a dynamic programming algorithm that solves the $\phi$-satisfiability problem for outerplanar graphs in $O(|V(G)|^3)$ time. This algorithm can also solve the $\eta$-defective 2-coloring problem for outerplanar graphs.