(Nearly-)tight bounds on the contiguity and linearity of cographs

In this paper we show that the contiguity and linearity of cographs on n   vertices are both O(logn). Moreover, we show that this bound is tight for contiguity as there exists a family of cographs on n   vertices whose contiguity is Ω(logn). We also provide an Ω(logn/loglogn) lower bound on the maximum linearity of cographs on n vertices. As a by-product of our proofs, we obtain a min–max theorem, which is worth of interest in itself, stating equality between the rank of a tree and the minimum height of one of its path partitions.