On the monophonic convexity in complementary prisms

Abstract

A set S of vertices of a graph G is monophonic convex if S contains all the vertices belonging to any induced path connecting two vertices of S. The cardinality of a maximum proper monophonically convex set of G is called the monophonic convexity number of G. The monophonic interval of a set S of vertices of G is the set S together with every vertex belonging to any induced path connecting two vertices of S. The cardinality of a minimum set S⊆V(G) whose monophonic interval is V(G) is called the monophonic interval number of G. The monophonic convex hull of a set S of vertices of G is the smallest monophonically convex set containing S in G. The cardinality of a minimum set S⊆V(G) whose monophonic convex hull is V(G) is called the monophonic hull number of G. The complementary prismGG¯ of G is obtained from the disjoint union of G and its complement G¯ by adding the edges of a perfect matching between them. In this work, we show that the corresponding decision problem to the monophonic convexity number is NP-complete even for complementary prisms. On the other hand, we show that the monophonic interval number and the monophonic hull number can be determined in linear time for the complementary prisms of all graphs.