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.
Highlights
Can we compute a maximum matching in a graph in linear-time? – i.e., computing a maximum set of pairwise disjoint edges in a graph. – Despite considerable years of research and the design of elegant combinatorial and linear programming techniques, the best-knownLeibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 6:2Pruned modular decomposition and Maximum Matching√ algorithms for this fundamental problem have stayed blocked to an O(m n)-time complexity on n-vertex m-edge graphs [22]
One appealing aspect of our approach in [9] was that, for most problems studied, we obtained a linear-time reduction from the input graph G to some quotient subgraph G in its modular decomposition. – We say that the problem is preserved by quotient. – This paved the way to the design of efficient algorithms for these problems on graph classes with unbounded modular-width, assuming their quotient subgraphs are simple enough w.r.t. the problem at hands
We propose pruning rules on the modules in a graph that can be used in order to compute Maximum Matching in linear-time on several new graph classes
Summary
Can we compute a maximum matching in a graph in linear-time? – i.e., computing a maximum set of pairwise disjoint edges in a graph. – Despite considerable years of research and the design of elegant combinatorial and linear programming techniques, the best-known. Can we compute a maximum matching in a graph in linear-time? Our work combines two successful approaches for this problem, namely, the use of a vertex-ordering characterization for certain graph classes [5, 10, 21], and a recent technique based on the decomposition of a graph by its modules [9]. We detail these two approaches in what follows, before summarizing our contributions
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