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Exploring Dominating Functions and Their Complexity in Subclasses of Weighted Chordal Graphs and Bipartite Graphs

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  • Chuan-Min Lee

    (Department of Applied Artificial Intelligence, Ming Chuan University, 5 De Ming Road, Guishan District, Taoyuan City 333, Taiwan)

Abstract

Domination problems are fundamental problems in graph theory with diverse applications in optimization, network design, and computational complexity. This paper investigates { k } -domination, k -tuple domination, and their total domination variants in weighted strongly chordal graphs and chordal bipartite graphs. Specifically, the { k } -domination problem in weighted strongly chordal graphs and the total { k } -domination problem in weighted chordal bipartite graphs are shown to be solvable in O ( n + m ) time. For weighted proper interval graphs and convex bipartite graphs, we solve the k -tuple domination and total k -tuple domination problems in O ( n 2.371552 log 2 ( n ) log ( n / δ ) ) , where δ is the desired accuracy. Furthermore, for weighted unit interval graphs, the k -tuple domination problem achieves a significant complexity improvement, reduced from O ( n k + 2 ) to O ( n 2.371552 log 2 ( n ) log ( n / δ ) ) . These results are achieved through a combination of linear and integer programming techniques, complemented by totally balanced matrices, totally unimodular matrices, and graph-specific matrix representations such as neighborhood and closed neighborhood matrices.

Suggested Citation

  • Chuan-Min Lee, 2025. "Exploring Dominating Functions and Their Complexity in Subclasses of Weighted Chordal Graphs and Bipartite Graphs," Mathematics, MDPI, vol. 13(3), pages 1-30, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:403-:d:1577192
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    References listed on IDEAS

    as
    1. Chun-Hung Liu & Gerard J. Chang, 2013. "Roman domination on strongly chordal graphs," Journal of Combinatorial Optimization, Springer, vol. 26(3), pages 608-619, October.
    2. Peng Li & Aifa Wang & Jianhui Shang, 2023. "A simple optimal algorithm for k-tuple dominating problem in interval graphs," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-9, January.
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