Separating the edges of a graph by cycles and by subdivisions of $K_4$

Abstract

A separating system of a graph $G$ is a family $\mathcal{S}$ of subgraphs of $G$ for which the following holds: for all distinct edges $e$ and $f$ of $G$, there exists an element in $\mathcal{S}$ that contains $e$ but not $f$. Recently, it has been shown that every graph of order $n$ admits a separating system consisting of $19n$ paths [Bonamy, Botler, Dross, Naia, Skokan, Separating the Edges of a Graph by a Linear Number of Paths, Adv. Comb., October 2023], improving the previous almost linear bound of $\mathrm{O}(n\log^\star n)$ [S. Letzter, Separating paths systems of almost linear size, Trans. Amer. Math. Soc., to appear], and settling conjectures posed by Balogh, Csaba, Martin, and Pluh\'ar and by Falgas-Ravry, Kittipassorn, Kor\'andi, Letzter, and Narayanan. We investigate a natural generalization of these results to subdivisions of cliques, showing that every graph admits both a separating system consisting of $41n$ edges and cycles, and a separating system consisting of $82 n$ edges and subdivisions of $K_4$.