A note on the convexity number for complementary prisms

TL;DR

This paper investigates the convexity number in the context of complementary prisms, proving NP-completeness of related decision problems, and providing exact values or bounds for specific classes of graphs.

Contribution

It establishes NP-completeness for the convexity number decision problem on complementary prisms and determines the convexity number for disconnected graphs and cographs, along with a lower bound for certain diameters.

Findings

NP-completeness of the convexity number decision problem for complementary prisms

Exact convexity number for disconnected graphs and cographs

Lower bound for graphs with diameter not equal to 3

Abstract

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.