Geodesic
A note on the convexity number for complementary prisms
Discrete mathematics & theoretical computer science, Tome 21 (2019) no. 4

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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 $diam(G) \neq 3$. 
@article{DMTCS_2019_21_4_a1,
     author = {Castonguay, Diane and Coelho, Erika M. M. and Coelho, Hebert and Nascimento, Julliano R.},
     title = {A note on the convexity number for complementary prisms},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2019},
     doi = {10.23638/DMTCS-21-4-4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-4-4/}
}
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Castonguay, Diane; Coelho, Erika M. M.; Coelho, Hebert; Nascimento, Julliano R. A note on the convexity number for complementary prisms. Discrete mathematics & theoretical computer science, Tome 21 (2019) no. 4. doi: 10.23638/DMTCS-21-4-4

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