On the monophonic convexity in complementary prisms

Abstract

A set $S$ of vertices of a graph $G$ is \emph{monophonic convex} if $S$ contains all the vertices belonging to any induced path connecting two vertices of $S$. The cardinality of a maximum proper monophonic convex set of $G$ is called the \emph{monophonic convexity number} of $G$. The \emph{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 $S \subseteq V(G)$ whose monophonic interval is $V(G)$ is called the \emph{monophonic number} of $G$. The \emph{monophonic convex hull} of a set $S$ of vertices of $G$ is the smallest monophonic convex set containing $S$ in $G$. The cardinality of a minimum set $S \subseteq V(G)$ whose monophonic convex hull is $V(G)$ is called the \emph{monophonic hull number} of $G$. The \emph{complementary prism} $\GG$ of $G$ is obtained from the disjoint union of $G$ and its complement $\overline{G}$ by adding the edges of a perfect matching between them. In this work, we determine the monophonic convexity number, the monophonic number, and the monophonic hull number of the complementary prisms of all graphs.