A representation for the modular-pairs of a cograph by modular decomposition

A modular-pair in a graph G is a pair { Q 1 , Q 2 } of disjoint sets of vertices in G such that each vertex of v ∈ V ( G ) \ ( Q 1 ∪ Q 2 ) is adjacent either to all vertices of Qj or none of them, for j = 1 , 2 , | Q 1 | ≥ 1 and | Q 2 | ≥ 1 . The modular-pair is a natural extension of the modules of a graph. In this paper we present a way to represent compactly all modular-pairs of a cograph G by using the modular decomposition of this cograph.