Linearity is strictly more powerful than contiguity for encoding graphs

Linearity and contiguity are two parameters devoted to graph encoding. Linearity is a generalization of contiguity in the sense that every encoding achieving contiguity k induces an encoding achieving linearity k, both encoding having size Theta(k.n), where n is the number of vertices of G. In this paper, we prove that linearity is a strictly more powerful encoding than contiguity, i.e. there exists some graph family such that the linearity is asymptotically negligible in front of the contiguity. We prove this by answering an open question asking for the worst case linearity of a cograph on n vertices: we provide an O(log n/log log n) upper bound which matches the previously known lower bound. (C) 2016 Elsevier B.V. All rights reserved.Region Rhone-Alpes, CNRS, Vietnam Institute for Advanced Study in Mathematics (VIASM), Vietnam National Foundation for Science and Technology Development (NAFOSTED), Fondecyt, Nucleo Milenio Informacion y Coordinacion en Redes (ACGO)