发表机构
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS; University of Johannesburg; Lebanese American University; Centrale Méditerranée, Laboratoire d’Informatique et Systèmes UMR 7020; University of Turku（克莱蒙奥弗涅大学; 约翰内斯堡大学; 黎巴嫩美国大学; 地中海中央理工学院; 图尔库大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明无孤立点图的顶点集可划分为一个支配集和一个定位支配集，由此推出支配数与定位支配数之和不超过顶点数且界紧，并给出多项式时间构造算法。
AI 中文摘要
设$G$是一个图，$S$是$G$的顶点集的一个子集，如果每个不在$S$中的顶点在$S$中都有一个邻居（两个顶点相邻则互为邻居），则称$S$为$G$的一个支配集。$G$的支配数$\gamma(G)$是$G$的所有支配集中的最小基数。给定图$G$的顶点集的一个子集$S$，如果两个顶点在$S$中的邻居集合不同，则称它们被$S$定位。此外，如果$S$定位每一对不在$S$中的顶点，则称$S$为$G$的一个定位集。$G$的一个定位支配集既是$G$的支配集又是$G$的定位集。定位支配数$\gamma^{\rm LD}(G)$是$G$的所有定位支配集中的最小基数。在定位支配集的研究中，一个著名的猜想是：对于阶为$n$的无孤立点且无孪生点的图，其定位支配数至多为$\frac{1}{2}n$。到目前为止，这个上界猜想的最佳近似已知为$\left \lceil \frac{5}{8}n \right \rceil$。与该猜想高度一致的是，文献中提出了一个更强的重新表述：是否可以将无孤立点且无孪生点的图的顶点集划分为两个定位集。然而，如果图允许有孪生点，这样的定位集划分可能不存在。沿着这一研究方向，我们证明：如果$G$是一个无孤立点（且不一定无孪生点）的图，那么$G$的顶点集可以被划分为一个支配集和一个定位支配集。作为推论，我们得出每个阶为$n$的无孤立点图$G$满足$\gamma(G) + \gamma^{\rm LD}(G) \le n$，并且我们证明这个界是紧的。此外，我们关于无孤立点图的顶点集可划分为一个支配集和一个定位支配集的证明还提供了一个构造这种划分的多项式时间算法。
英文摘要
A set $S$ of vertices in a graph $G$ is a dominating set of $G$ if every vertex not in $S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. The domination number, $γ(G)$, of $G$ is the minimum cardinality among all dominating sets of $G$. Given a set $S$ of vertices of a graph $G$, two vertices are located by $S$ if they have distinct sets of neighbors in $S$. Moreover, if $S$ locates every pair of vertices not in $S$, then it is called a locating set of $G$. A locating dominating set of $G$ is both a dominating and a locating set of $G$. The locating domination number, $γ^{\rm LD}(G)$, is the minimum cardinality among all locating dominating sets of $G$. A notable conjecture in the study of locating dominating sets is to show that the locating domination number of an isolate-free and twin-free graph of order $n$ is at most $\frac{1}{2}n$. So far, the best approximation to this upper bound conjecture is known to be $\left \lceil \frac{5}{8}n \right \rceil$. Much in line with the conjecture, an even stronger reformulation proposed in the literature asks if it is possible to partition the vertex set of an isolate-free and twin-free graph into two locating sets. However, such partitions into locating sets may not exist if the graph is also allowed to have twins. Continuing with this line of research, we show that if $G$ is an isolate-free (and not necessarily twin-free) graph, then the vertex set of $G$ can be partitioned into a dominating set and a locating dominating set. As a consequence, we infer that every isolate-free graph $G$ of order $n$ satisfies $γ(G) + γ^{\rm LD}(G) \le n$, and we show that the last bound is tight. Moreover, our proof of the existence of a partition of the vertex set of an isolate-free graph into a dominating and a locating dominating set also provides a polynomial-time algorithm to construct such a partition.
Journal refThe Electronic Journal of Combinatorics 33(1): #P1.51 (2026)
DOI：10.37236/14049
