Openindependent, openlocatingdominating sets: structural aspects of some classes of graphs
Abstract
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{opendominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{openindependent, openlocatingdominating} (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 opendominated 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 DOI:
 10.48550/arXiv.2103.07972
 arXiv:
 arXiv:2103.07972
 Bibcode:
 2021arXiv210307972C
 Keywords:

 Computer Science  Discrete Mathematics
 EPrint:
 18 pages, 5 figures