Dominating Induced Matchings for P7-free Graphs in Linear Time

• Published in: Algorithms and Computation pp. 100 - 109
• Main Authors: Brandstädt, Andreas, Mosca, Raffaele
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: dominating induced matching; efficient edge domination; free graphs; linear time algorithm; robust algorithm
• ISBN: 9783642255908; 3642255906
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-25591-5_12

Let G be a finite undirected graph with edge set E. An edge set E′ ⊆ E is an induced matching in G if the pairwise distance of the edges of E′ in G is at least two; E′ is dominating in G if every edge e ∈ E ∖ E′ intersects some edge in E′. The Dominating Induced Matching Problem (DIM, for short) asks for the existence of an induced matching E′ which is also dominating in G; this problem is also known as the Efficient Edge Domination Problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{NP}$\end{document}-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for Pk-free graphs for any k ≥ 5; Pk denotes a chordless path with k vertices and k − 1 edges. We show in this paper that the weighted DIM problem is solvable in linear time for P7-free graphs in a robust way.