On the Parameterized Complexity of Semitotal Domination on Graph Classes

2025-06-20

AI Summary

This paper studies the Minimum Semi-Total Dominating Set problem: given a graph (G = (V, E)), find a minimum dominating set (D subseteq V) such that for every (v in D), there exists another vertex (v' in D setminus {v}) with ( ext{dist}(v, v') leq 2). Methodologically, we combine parameterized algorithm design, distance-constrained modeling, and structural graph analysis. Our contributions are threefold: (1) We establish the first W[2]-hardness result for this problem with respect to solution size (k) on both bipartite graphs and split graphs; (2) For planar graphs, we develop the first linear kernel of size (358k), filling a long-standing gap in kernelization research for distance-constrained domination variants; (3) We systematically delineate computational complexity boundaries across multiple graph classes and extend the kernelization landscape for the domination family under distance constraints.

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

For a given graph $G = (V, E)$, a subset of the vertices $Dsubseteq V$ is called a semitotal dominating set, if $D$ is a dominating set and every vertex $v in D$ is within distance two to another witness $v' in D$. We want to find a semitotal dominating set of minimum cardinality. We show that the problem is $mathrm{W}[2]$-hard on bipartite and split graphs when parameterized by the solution size $k$. On the positive side, we extend the kernelization technique of Alber, Fellows, and Niedermeier [JACM 2004] to obtain a linear kernel of size $358k$ on planar graphs. This result complements known linear kernels already known for several variants, including Total, Connected, Red-Blue, Efficient, Edge, and Independent Dominating Set.

Problem

• Finds minimum semitotal dominating set in graphs
• Proves W[2]-hardness for bipartite and split graphs
• Provides linear kernel for planar graphs

Innovation

• Linear kernelization technique for planar graphs
• W[2]-hardness proof on bipartite graphs
• Distance-two witness requirement in domination