On the monophonic convexity in complementary prisms

TL;DR

This paper investigates the properties of monophonic convexity in complementary prisms of graphs, specifically determining key convexity parameters for all such graphs.

Contribution

It provides the first comprehensive analysis of monophonic convexity parameters in complementary prisms of arbitrary graphs.

Findings

Calculated the monophonic convexity number for all complementary prisms.

Determined the monophonic number and hull number for these structures.

Established formulas and bounds for these parameters in the context of 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.