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 $K_4$.

翻译：图$G$的一个分离系统是指$G$的子图族$\mathcal{S}$，其满足以下性质：对于$G$中任意两条不同的边$e$和$f$，均存在$\mathcal{S}$中的一个元素包含$e$但不包含$f$。最近的研究表明，每个阶数为$n$的图都存在一个由$19n$条路径构成的分离系统[Bonamy, Botler, Dross, Naia, Skokan, 《用线性数量的路径分离图的边》，Adv. Comb., 2023年10月]，该结果改进了先前$\mathrm{O}(n\log^\star n)$的近似线性界[S. Letzter, 《具有近似线性规模的路径分离系统》，Trans. Amer. Math. Soc., 即将出版]，并解决了Balogh、Csaba、Martin、Pluhár以及Falgas-Ravry、Kittipassorn、Korándi、Letzter、Narayanan提出的猜想。我们研究这些结果向团细分结构的自然推广，证明每个图都存在一个由$41n$条边和环构成的分离系统，以及一个由$82n$条边和$K_4$细分结构构成的分离系统。