GEODETIC GAMES FOR GRAPHS

Abstract

Let S be a subset of the vertex set V(G) of a nontrivial connected graph G. The geodetic closure (S) of S is the set of all vertices on geodesics between two vertices in S. The first player A chooses a vertex v₁ of G. The second player B then picks v₂ ≠ v₁ and forms the geodetic closure (S₂) = ({v₁, v₂}). Now A selects v₃ ε V—S₂ and forms (S₃) = ({v₁, v₂, v₃}), etc. The player who first selects a vertex v_n such that (S_n) = V wins the achievement game, but loses the avoidance game. These geodetic achievement and avoidance games are solved for several families of graphs by determining which player is the winner.