Locating-dominating partitions for some classes of graphs
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2025
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| _version_ | 1866910133559230464 |
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| author | Foucaud, Florent Maniya, Paras Vinubhai Paul, Kaustav Pradhan, Dinabandhu |
| author_facet | Foucaud, Florent Maniya, Paras Vinubhai Paul, Kaustav Pradhan, Dinabandhu |
| contents | A dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G) \setminus D$ is adjacent to at least one vertex in $D$. A set $L\subseteq V(G)$ is a locating set of $G$ if every vertex in $V(G) \setminus L$ has pairwise distinct open neighborhoods in $L$. A set $D\subseteq V(G)$ is a locating-dominating set of $G$ if $D$ is a dominating set and a locating set of $G$. The location-domination number of $G$, denoted by $γ_{LD}(G)$, is the minimum cardinality among all locating-dominating sets of $G$. A well-known conjecture in the study of locating-dominating sets is that if $G$ is an isolate-free and twin-free graph of order $n$, then $γ_{LD}(G)\le \frac{n}{2}$. Recently, Bousquet et al. [Discrete Math. 348 (2025), 114297] proved that if $G$ is an isolate-free and twin-free graph of order $n$, then $γ_{LD}(G)\le \lceil\frac{5n}{8}\rceil$ and posed the question whether the vertex set of such a graph can be partitioned into two locating sets. We answer this question affirmatively for twin-free distance-hereditary graphs, maximal outerplanar graphs, split graphs, and co-bipartite graphs. In fact, we prove a stronger result that for any graph $G$ without isolated vertices and twin vertices, if $G$ is a distance-hereditary graph or a maximal outerplanar graph or a split graph or a co-bipartite graph, then the vertex set of $G$ can be partitioned into two locating-dominating sets. Consequently, this also confirms the original conjecture for these graph classes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Locating-dominating partitions for some classes of graphs Foucaud, Florent Maniya, Paras Vinubhai Paul, Kaustav Pradhan, Dinabandhu Combinatorics Discrete Mathematics 05C69 A dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G) \setminus D$ is adjacent to at least one vertex in $D$. A set $L\subseteq V(G)$ is a locating set of $G$ if every vertex in $V(G) \setminus L$ has pairwise distinct open neighborhoods in $L$. A set $D\subseteq V(G)$ is a locating-dominating set of $G$ if $D$ is a dominating set and a locating set of $G$. The location-domination number of $G$, denoted by $γ_{LD}(G)$, is the minimum cardinality among all locating-dominating sets of $G$. A well-known conjecture in the study of locating-dominating sets is that if $G$ is an isolate-free and twin-free graph of order $n$, then $γ_{LD}(G)\le \frac{n}{2}$. Recently, Bousquet et al. [Discrete Math. 348 (2025), 114297] proved that if $G$ is an isolate-free and twin-free graph of order $n$, then $γ_{LD}(G)\le \lceil\frac{5n}{8}\rceil$ and posed the question whether the vertex set of such a graph can be partitioned into two locating sets. We answer this question affirmatively for twin-free distance-hereditary graphs, maximal outerplanar graphs, split graphs, and co-bipartite graphs. In fact, we prove a stronger result that for any graph $G$ without isolated vertices and twin vertices, if $G$ is a distance-hereditary graph or a maximal outerplanar graph or a split graph or a co-bipartite graph, then the vertex set of $G$ can be partitioned into two locating-dominating sets. Consequently, this also confirms the original conjecture for these graph classes. |
| title | Locating-dominating partitions for some classes of graphs |
| topic | Combinatorics Discrete Mathematics 05C69 |
| url | https://arxiv.org/abs/2506.12933 |