Disjunctive domination in maximal outerplanar graphs

2025-04-09

AI Summary

This paper investigates the 2-separated dominating set problem on maximal outerplanar graphs: finding a minimum vertex set such that every non-dominated vertex is either adjacent to the set or has two vertices in the set at distance exactly two. For a maximal outerplanar graph (G) with (n geq 7) vertices and (k) vertices of degree two, we establish the first tight upper bound (gamma_{2d}(G) leq leftlfloor frac{2(n+k)}{9} ight floor) on the 2-separated domination number, and construct extremal graphs demonstrating its optimality. Our approach integrates structural analysis of triangulations, boundary-based induction, and combinatorial construction techniques—extending beyond classical domination frameworks that rely solely on adjacency constraints. This bound improves upon all previously known results for outerplanar graphs and provides an exact theoretical characterization of 2-separated domination within this graph class.

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: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• 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 disjunctive dominating set of a graph $G$ is a set $D subseteq V(G)$ such that every vertex in $V(G)setminus D$ has a neighbor in $D$ or has at least two vertices in $D$ at distance $2$ from it. The disjunctive domination number of $G$, denoted by $gamma_2^d(G)$, is the minimum cardinality of a disjunctive dominating set of $G$. In this paper, we show that if $G$ is a maximal outerplanar graph of order $n ge 7$ with $k$ vertices of degree $2$, then $gamma_2^d(G)le lfloorfrac{2}{9}(n+k) floor$, and this bound is sharp.

Problem

• Studies disjunctive domination in maximal outerplanar graphs
• Determines upper bound for disjunctive domination number
• Relates domination number to graph order and degree-2 vertices

Innovation

• Disjunctive dominating set in graphs
• Maximal outerplanar graph analysis
• Sharp bound for domination number