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The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes

Authors Guillaume Ducoffe, Alexandru Popa



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Guillaume Ducoffe
  • ICI – National Institute for Research and Development in Informatics, Bucharest, Romania , The Research Institute of the University of Bucharest ICUB, Bucharest, Romania
Alexandru Popa
  • University of Bucharest, Bucharest, Romania , ICI – National Institute for Research and Development in Informatics, Bucharest, Romania

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Guillaume Ducoffe and Alexandru Popa. The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 6:1-6:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ISAAC.2018.6

Abstract

We address the following general question: given a graph class C on which we can solve Maximum Matching in (quasi) linear time, does the same hold true for the class of graphs that can be modularly decomposed into C? As a way to answer this question for distance-hereditary graphs and some other superclasses of cographs, we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. We remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a "pruned modular decomposition", that can be computed in quasi linear time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. The latter result strictly extends a recent framework in (Coudert et al., SODA'18). Our work is the first to explain why the existence of some nice ordering over the modules of a graph, instead of just over its vertices, can help to speed up the computation of maximum matchings on some graph classes.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Design and analysis of algorithms
Keywords
  • maximum matching
  • FPT in P
  • modular decomposition
  • pruned graphs
  • one-vertex extensions
  • P_4-structure

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