Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

TL;DR

This paper investigates the structural properties of certain graph classes related to open-independent, open-locating-dominating sets, proving NP-completeness for various graph classes and characterizing specific graph subclasses with these sets.

Contribution

It establishes the NP-completeness of the OLD_oind-set problem for several graph classes and provides characterizations for P_4-tidy graphs and complementary prisms of cographs.

Findings

NP-complete for planar bipartite graphs of max degree five and girth six

NP-complete for planar subcubic graphs of girth nine

Characterizations of P_4-tidy graphs and cograph prisms with OLD_oind-sets

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{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.