Common connected components of interval graphs

The Common Connected Problem (CCP) consists in identifying common connected components in two or more graphs on the same vertices (or reduced to). More formally, let G1(V,E1) and G2(V,E2) be two such graphs and let V ′ ⊂ V . If G1[V ] and G2[V ] are both connected, V ′ is said a common connected component. The CCP problem is the identification of maximal (for the inclusion order) such components, that form a partition of V . Let n = |V | and m = |E1| + |E2|. We present an O((n + m) logn) worst case time algorithm solving the CCP problem when G1 and G2 are two interval graphs. The algorithm combines maximal clique path decompositions of the two input graphs together with an Hopcroft like partitioning approach.