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

Authors:
Christophe Crespelle;Philippe Gambette
Affiliations:
Université Claude Bernard Lyon 1, DANTE/INRIA, LIP UMR CNRS 5668, ENS de Lyon, Université de Lyon, France;Université Paris-Est, LIGM UMR CNRS 8049, Université Paris-Est Marne-la-Vallée, 5 boulevard Descartes, 77420 Champs-sur-Marne, France
Venue:
Theoretical Computer Science
Year:
2014

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Abstract

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 @W(logn). We also provide an @W(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.