Computing H-Joins with Application to 2-Modular Decomposition

We present here a general framework to design algorithms that compute H-join. For a given bipartite graph H, we say that a graph G admits a H-join decomposition or simply a H-join, if the vertices of G can be partitioned in |H| parts connected as in H. This graph H is a kind of pattern, that we want to discover in G. This framework allows us to present fastest known algorithms for the computation of P ₄-join (aka N-join), P ₅-join (aka W-join), C ₆-join (aka 6-join). We also generalize this method to find a homogeneous pair (also known as 2-module), a pair {M ₁,M ₂} such that for every vertex x?(M ₁∪M ₂) and i∈{1,2}, x is either adjacent to all vertices in M _i or to none of them. First used in the context of perfect graphs (Chvátal and Sbihi in Graphs Comb. 3:127–139, 1987), it is a generalization of splits (a.k.a. 1-joins) and of modules. The algorithmics to compute them appears quite involved. In this paper, we describe an O(mn ²)-time algorithm computing all maximal homogeneous pairs of a graph, which not only improves a previous bound of O(mn ³) for finding only one pair (Everett et al. in Discrete Appl. Math. 72:209–218, 1997), but also uses a nice structural property of homogenous pairs, allowing to compute a canonical decomposition tree for sesquiprime graphs (i.e., graphs G having no module and such that for every vertex v∈G, G?v also has no module).