Complexity results related to monophonic convexity - Dourado, Mitre C.; Protti, Fábio; Szwarcfiter, Jayme L. (Discrete Applied Mathematics, 2010)
The study of monophonic convexity is based on the family of induced paths of a graph. The closure of a subset X of vertices, in this case, contains every vertex v such that v belongs to some induced path linking two vertices of X . Such a closure is called monophonic closure. Likewise, the convex hull of a subset is called monophonic convex hull. In this work we deal with the computational complexity of determining important convexity parameters, considered in the context of monophonic convexity. Given a graph G , we focus on three parameters: the size of a maximum proper convex subset of G ( m-convexity number); the size of a minimum subset whose closure is equal to V ( G ) ( monophonic number); and the size of a minimum subset whose convex hull is equal to V ( G ) ( m-hull number). We prove that the decision problems corresponding to the m-convexity and monophonic numbers are NP-complete, and we describe a polynomial time algorithm for computing the m-hull number of an arbitrary graph.