The use of a pruned modular decomposition for MAXIMUM MATCHING algorithms on some graph classes

Guillaume Ducoffe, Alexandru Popa

Research output: Contribution to journalArticlepeer-review

Abstract

Consider the following general question: if we can solve MAXIMUM MATCHING in (quasi) linear time on a graph class C, does the same hold true for the class of graphs that can be modularly decomposed into C ? What makes the latter question difficult is that the MAXIMUM MATCHING problem is not preserved by quotient, thereby making difficult to exploit the structural properties of the quotient subgraphs of the modular decomposition. So far, we are only aware of a recent framework in (Coudert et al., SODA’18) that only applies when the quotient subgraphs have bounded order and/or under additional assumptions on the nontrivial modules in the graph. As an attempt 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. Specifically, 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 the framework of Coudert et al. Our work is the first to use some ordering over the modules of a graph, instead of just over its vertices, in order to speed up the computation of maximum matchings on some graph classes.

Original languageEnglish
Pages (from-to)201-222
Number of pages22
JournalDiscrete Applied Mathematics
Volume291
DOIs
Publication statusPublished - Mar 11 2021

Keywords

  • FPT in P
  • Maximum matching
  • Modular decomposition
  • One-vertex extensions
  • P-structure
  • Pruned graphs

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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