## Castonguay, Diane and Coelho, Erika M. M. and Coelho, Hebert and Nascimento, Julliano R. - A note on the convexity number for complementary prisms

dmtcs:4853 - Discrete Mathematics & Theoretical Computer Science, July 30, 2019, vol. 21 no. 4 - https://doi.org/10.23638/DMTCS-21-4-4
A note on the convexity number for complementary prisms

Authors: Castonguay, Diane and Coelho, Erika M. M. and Coelho, Hebert and Nascimento, Julliano R.

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 $diam(G) \neq 3$.

Volume: vol. 21 no. 4
Section: Graph Theory
Published on: July 30, 2019
Submitted on: September 25, 2018
Keywords: Computer Science - Discrete Mathematics