TY - GEN

T1 - The use of a pruned modular decomposition for maximum matching algorithms on some graph classes

AU - Ducoffe, Guillaume

AU - Popa, Alexandru

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - FPT in P

KW - Maximum matching

KW - Modular decomposition

KW - One-vertex extensions

KW - P -structure

KW - Pruned graphs

UR - http://www.scopus.com/inward/record.url?scp=85063667696&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063667696&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2018.6

DO - 10.4230/LIPIcs.ISAAC.2018.6

M3 - Conference contribution

AN - SCOPUS:85063667696

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 29th International Symposium on Algorithms and Computation, ISAAC 2018

A2 - Lee, Der-Tsai

A2 - Liao, Chung-Shou

A2 - Hsu, Wen-Lian

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 29th International Symposium on Algorithms and Computation, ISAAC 2018

Y2 - 16 December 2018 through 19 December 2018

ER -