The use of a pruned modular decomposition for Maximum Matching algorithms on some graph classes

Consider the following general question: if we can solve Maximum Matching in (quasi) linear time on a graph class C , does same hold true for of graphs that be modularly decomposed into ? What makes latter question difficult is problem not preserved by quotient, thereby making to exploit structural properties quotient subgraphs modular decomposition. So far, are only aware recent framework (Coudert et al., SODA’18) applies when have bounded order and/or under additional assumptions nontrivial modules graph. As an attempt answer this distance-hereditary and some other superclasses cographs, study combined effect decomposition with pruning process over subgraphs. Specifically, remove sequentially from all such their so-called one-vertex extensions ( i.e. pendant, anti-pendant, twin, universal isolated vertices). Doing so, obtain “pruned decomposition”, computed quasi time. Our main result pruned then maximum matching The strictly extends Coudert al. work first use ordering graph, instead just its vertices, speed up computation matchings classes.