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.
Highlights
IntroductionConsider the situation where a graph G models a facility or a multiprocessor network with limited-range detection devices (sensing for example, movement, heat or size) that are placed at chosen vertices of G
Consider the situation where a graph G models a facility or a multiprocessor network with limited-range detection devices that are placed at chosen vertices of G
As the problem is known to be N P-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 for planar subcubic graphs of girth nine
Summary
Consider the situation where a graph G models a facility or a multiprocessor network with limited-range detection devices (sensing for example, movement, heat or size) that are placed at chosen vertices of G. In this paper we consider the following more restrictive problem: OLD-OIND (existence of an open-independent, open locating dominating set) Instance: A graph G. Complementary prisms are a class of apparently well-behaved graphs, many N P-complete problems for general graphs remain N Pcomplete for this class, for example, finding an independent or a dominating set, or establishing P3-convexity [9]. We study some graph classes for which the problem can be solved in polynomial time and present characterizations of both P4-tidy graphs and complementary prisms of cographs that have an OLDoind-set. If an open-independent, open-locating-dominating set (an OLDoind-set for short) exists in a given graph G, it is often of interest to establish a set of minimum size among such sets in G, which is denoted by OLDoind(G). For a set X ⊆ V (G), let X denote the corresponding vertices of X in V (G)
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