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 and also for planar subcubic graphs. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.
翻译:Let G=( V( G), E( G) $ 是一个限定的简单不方向的图表, 顶点设置为 $V( G) 美元, 顶点设定为 $E( G) 美元, 顶点设定为 $S\ subseq V( G) 美元。 如果每根G$的顶点至少有一个以美元为单位的相邻者, 而每根G$的顶点至少有一个以美元为单位的相邻者, 和 emph{ 开放独立、 开放分配 - 支配} ( $OLDççççoond) 美元设定为短期), 如果$G 的两张顶点没有以美元为单位的相邻者, 而每张S$的顶点完全以美元为主 。 要决定$G$是否有一个美元为单位的顶点是否以美元为单位的顶点, 有在别处报告的重要应用程序。 众所周知, 问题是$\math dald=Dc_ PNP}, 似乎很困难,因为我们显示其复杂性仍然相同的平面图图。