Open-separating dominating codes in graphs

2024-02-05

International Symposium on Combinatorial Optimization

AI Summary

This paper introduces the open separable dominating code (OSD-code)—a novel dominating set requiring that each vertex’s open neighborhood intersects the code in a unique subset. The study investigates the existence, minimum cardinality, and computational complexity of OSD-codes. Methodologically, the authors first formally define OSD-codes, establish a comparative framework with open locating-dominating codes (OLD-codes), and reformulate the problem as a hypergraph covering problem; they then employ graph-theoretic analysis, integer programming, and polyhedral combinatorics to derive existence conditions and tight bounds for trees, interval graphs, and other graph classes. Results show that OSD-code recognition is NP-complete, and the associated covering polytope admits a characterizable minimal support structure. The primary contribution lies in introducing and systematically formalizing “open-neighborhood separation” as a new domination paradigm, thereby enriching the theoretical foundations of combinatorial domination.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Application Domains: Humanities & Computational Social Science

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is to choose a suitable dominating set $C$ of a graph $G$ such that the neighbourhoods of all vertices of $G$ have distinct intersections with $C$. Such a dominating and separating set $C$ is often referred to as a emph{code} in the literature. Depending on the types of dominating and separating sets used, various problems arise under various names in the literature. In this paper, we introduce a new problem in the same realm of identification problems whereby the code, called emph{open-separating dominating code}, or emph{OSD-code} for short, is a dominating set and uses open neighbourhoods for separating vertices. The paper studies the fundamental properties concerning the existence, hardness and minimality of OSD-codes. Due to the emergence of a close and yet difficult to establish relation of the OSD-codes with another well-studied code in the literature called open locating dominating codes, or OLD-codes for short, we compare the two on various graph families. Finally, we also provide an equivalent reformulation of the problem of finding OSD-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with OSD-codes, again in relation to OLD-codes of some graph families already studied in this context.

Problem

• Introducing open-separating dominating codes for vertex identification
• Studying existence, hardness, and minimality of OD-codes in graphs
• Comparing OD-codes with OTD-codes across different graph families

Innovation

• Open-separating dominating code for vertex identification
• Dominating set with open neighbourhoods separation
• Hypergraph covering reformulation for code discovery