Separating Path Systems of Size at most $7.75n$

Abstract

A family of paths in a graph $G$ strongly separates the edges of $G$ if for every ordered pair of distinct edges $(e,f)$ some path in the family contains $e$ and avoids $f$; the minimum size of such a family is denoted by $\mathrm{ssp}(G)$. Bonamy, Botler, Dross, Naia, and Skokan proved, for every $n$-vertex graph $G$, that $\mathrm{ssp}(G)\le 19n$; Liu, Xu, and Yang recently improved this to $\mathrm{ssp}(G)\le 10n-o(n)$. We prove, for every $n$-vertex graph $G$, that $\mathrm{ssp}(G)\le 7.75n$.