Linearity is Strictly More Powerful than Contiguity for Encoding Graphs

TL;DR

This paper demonstrates that linearity is a strictly more powerful graph encoding than contiguity by establishing an asymptotic separation and providing tight bounds for cographs.

Contribution

It proves linearity's superiority over contiguity for graph encoding and answers an open question on the worst-case linearity of cographs with tight bounds.

Findings

Linearity is strictly more powerful than contiguity for graph encoding.

Established an $O(rac{ ext{log} n}{ ext{log} ext{log} n})$ upper bound for cograph linearity.

Matched the upper bound with a known lower bound for cographs.

Abstract

Linearity and contiguity are two parameters devoted to graph encoding. Linearity is a generalisation 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.