## Márcia R. Cappelle ; Erika Coelho ; Les R. Foulds ; Humberto J. Longo - Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

dmtcs:8440 - Discrete Mathematics & Theoretical Computer Science, February 7, 2022, vol. 24, no. 1 - https://doi.org/10.46298/dmtcs.8440
Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Authors: Márcia R. Cappelle ; Erika Coelho ; Les R. Foulds ; Humberto J. Longo

Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.

Volume: vol. 24, no. 1
Section: Graph Theory
Published on: February 7, 2022
Accepted on: January 5, 2022
Submitted on: September 2, 2021
Keywords: Computer Science - Discrete Mathematics