New Results on Vertices that Belong to Every Minimum Locating-Dominating Code

2025-09-01

AI Summary

This paper investigates structural and extremal properties of *min-forced vertices*—vertices necessarily contained in every minimum locating-dominating code of a graph. Method: We systematically introduce and characterize this notion via graph-theoretic constructions and combinatorial analysis, derive tight upper bounds on the number and density of min-forced vertices, determine the exact number of minimum locating-dominating codes in paths, and establish computational complexity via reduction. Results: We prove that the number of min-forced vertices is at most $frac{2}{3}(n - gamma^{LD}(G))$, and their proportion in any connected nontrivial graph is at most $2/5$; both bounds are tight. For path graphs, we completely determine the exact count of minimum locating-dominating codes. Moreover, we show that deciding whether a given vertex is min-forced is co-NP-hard. This work fills a fundamental gap in locating-dominating theory concerning forced structures and offers new insights into the interplay between code robustness and graph invariants.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Locating-dominating codes have been studied widely since their introduction in the 1980s by Slater and Rall. In this paper, we concentrate on vertices that must belong to all minimum locating-dominating codes in a graph. We call them min-forced vertices. We show that the number of min-forced vertices in a connected nontrivial graph of order $n$ is bounded above by $frac{2}{3}left(n -γ^{LD}(G) ight)$, where $γ^{LD}(G)$ denotes the cardinality of a minimum locating-dominating code. This implies that the maximum ratio between the number of min-forced vertices and the order of a connected nontrivial graph is at most $frac{2}{5}$. Moreover, both of these bounds can be attained. We also determine the number of different minimum locating-dominating codes in all paths. In addition, we show that deciding whether a vertex is min-forced is co-NP-hard.

Problem

• Characterizing vertices essential to all minimum locating-dominating codes
• Establishing upper bounds for min-forced vertices in graphs
• Proving computational complexity of min-forced vertex decision

Innovation

• Upper bound on min-forced vertices in graphs
• Maximum ratio calculation for connected graphs
• co-NP-hard decision for min-forced vertices