Convexity in graphs and hypergraphs
SIAM Journal on Algebraic and Discrete Methods
Discrete Mathematics
Convex sets in graphs, II. Minimal path convexity
Journal of Combinatorial Theory Series A
Chordless paths through three vertices
Theoretical Computer Science - Parameterized and exact computation
Graph Theory
On the hull number of some graph classes
Theoretical Computer Science
On the Carathéodory number of interval and graph convexities
Theoretical Computer Science
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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.