Convexity in Graphs: Achievement and Avoidance Games

Abstract

The convex hull of a set S of points of a graph G is the smallest set T containing S such that all the points in a geodesic joining two points of T lie in T . The convex hull T can also be formed by taking all geodesics joining two points of S and iterating that operation. The number of times this is done to S to get T is gin( S ), the geodetic iteration number of S . gin( G ) is defined as the maximum of gin( S ) over all sets S of points of G . The smallest number of points in a graph G such that gin( G ) = n was determined by Harary and Nieminen. In an achievement game, the first of two players to attain the stated goal wins; in an avoidance game, he/she loses. The chapter presents three types of achievement and avoidance games involving the convex hull, the geodetic hull, and geodesics from a point to a set of points; this leads to several unsolved problems.