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 $S$ of subgraphs of $G$ for which the following holds: for all distinct edges $e$ and $f$ of $G$, there exists an element in $S$ 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, improving the previous almost linear bound of $O(n{\log }^{\star } n)$, and settling conjectures posed by Balogh, Csaba, Martin, and Pluhár and by Falgas-Ravry, Kittipassorn, Korándi, 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 $82n$ edges and subdivisions of $K_4$.