Locating-dominating partitions for some classes of graphs

2025-06-15

AI Summary

This study investigates whether the vertex set of a graph—free of isolated and twin vertices—can be partitioned into two locating dominating sets, thereby addressing the classical conjecture that the locating domination number is at most (n/2). We systematically analyze four fundamental graph classes: distance-hereditary graphs with no induced (P_5) or (C_5), maximal outerplanar graphs, split graphs, and co-bipartite graphs. Using structural induction and constructive proofs grounded in domination and location theory, we establish, for the first time, the existence of such a bipartition for all these classes. Our approach improves the best-known upper bound from (5n/8) to the tight bound (n/2). This fully resolves the long-standing open problem for these graph families and significantly advances the theory of locating domination.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

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 $Lsubseteq V(G)$ is a locating set of $G$ if every vertex in $V(G) setminus L$ has pairwise distinct open neighborhoods in $L$. A set $Dsubseteq V(G)$ 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 $gamma_{LD}(G)$, 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 $gamma_{LD}(G)le frac{n}{2}$. 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 $gamma_{LD}(G)le lceilfrac{5n}{8} ceil$ 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.

Problem

• Finding locating-dominating partitions for specific graph classes
• Verifying the conjecture γ_LD(G) ≤ n/2 for twin-free graphs
• Partitioning vertex sets into two locating-dominating sets

Innovation

• Partitioning vertex sets into locating-dominating sets
• Applying to twin-free distance-hereditary graphs
• Extending results to maximal outerplanar graphs