Separating path systems for cubic graphs and for complete bipartite graphs

TL;DR

This paper investigates the strong separation number in various classes of graphs, establishing upper bounds for 2-degenerate, subcubic, and planar graphs, and deriving bounds for complete bipartite graphs with constructions matching some bounds.

Contribution

It introduces new bounds for the strong separation number in specific graph classes and provides constructions that achieve these bounds, advancing understanding of path systems in graphs.

Findings

Strong separation number of 2-degenerate graphs is at most n.

Upper bounds for subcubic, planar, and planar bipartite graphs.

Lower bounds for complete bipartite graphs with matching constructions.

Abstract

A strongly separating path system in a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that, for every two edges $e$ and $f$ of $G$, there is a paths in $\mathcal{P}$ with $e$ and not $f$, and vice-versa. The minimum number of such a system is the so called strong separation number of $G$. We prove that the strong separation number of every $2$-degenerate graph on $n$ vertices is at most $n$. Using this, we also provide upper bounds for the strong separation number of subcubic graphs, planar graphs, and planar bipartite graphs. On the other hand, we prove that the strong separation number a complete bipartite graph $K_{a,b}$ is at least $b$ if $a<b/2$ and at least $(\sqrt{6(b/2)+4}-2)a$ if $b/2\leq a\leq b$, and we provide a construction that attains the former bound.