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, has been studied elsewhere and is known to be $\mathcal{NP}$-complete. Reinforcing this result, it appears that the problem is notoriously difficult as we show that its complexity remains the same even for just planar bipartite 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, 上面设置为 $V( G) 美元, 边设定为 $E( G) 美元, 顶设定为 $S\ subseteq V( G) 美元。 $S 被称为 emph{ 开放支配 } 如果每个 G$ 的顶点至少有一个以美元为单位的邻居, 并且 \ emph{ 开放独立, 开放分配 - 支配} ( $OLDççççoond) 设定为短) 。 如果$G 中没有两只设定为 $S 的顶点, 边设定为 $( G) 和 $( $) 的顶点为 。 决定$G$( ) 是否有美元为美元或不是 开放 - 开放 - 开放, 开放 - 开放 - 开放 - 开放 - 分配 - 支配} ( 设置为 短 美元) ( 美元) ( $OL Dçoundn ) ) 设置为 设置为 。 问题似乎非常困难。