A note on module-composed graphs

TL;DR

This paper introduces module-composed graphs, characterizes their properties, and provides efficient algorithms for recognizing and constructing such graphs, revealing their relation to well-known graph classes like distance hereditary graphs.

Contribution

It defines module-composed graphs, proves their properties, and offers an O(n(m+n)) recognition algorithm, connecting them to bipartite distance hereditary graphs.

Findings

Module-composed graphs are HHDS-free and homogeneously orderable.

Every bipartite distance hereditary graph is module-composed.

Recognition algorithm runs in O(|V|(|V|+|E|)) time.

Abstract

In this paper we consider module-composed graphs, i.e. graphs which can be defined by a sequence of one-vertex insertions v_1,...,v_n, such that the neighbourhood of vertex v_i, 2<= i<= n, forms a module (a homogeneous set) of the graph defined by vertices v_1,..., v_{i-1}. We show that module-composed graphs are HHDS-free and thus homogeneously orderable, weakly chordal, and perfect. Every bipartite distance hereditary graph, every (co-2C_4,P_4)-free graph and thus every trivially perfect graph is module-composed. We give an O(|V_G|(|V_G|+|E_G|)) time algorithm to decide whether a given graph G is module-composed and construct a corresponding module-sequence. For the case of bipartite graphs, module-composed graphs are exactly distance hereditary graphs, which implies simple linear time algorithms for their recognition and construction of a corresponding module-sequence.