A Congestion Parameter for Depth-First Graph Traversals

Abstract

We explore a new graph parameter, the KLX number, which quantifies the minimum edge congestion of depth-first search (DFS) traversals of a given graph. Originally motivated by a problem in RNA nanostructure design, this parameter is also of independent theoretical interest. Informally, the KLX number of a graph is defined as the minimum, over all its DFS traversals, of the maximum number of back edges that are simultaneously open during the traversal. We provide full characterisations and linear-time recognition algorithms for graphs with KLX numbers 0, 1 and 2. We also relate KLX to tree-width, proving that any graph satisfies $\mathrm{TW}\le \mathrm{KLX}+1$. Furthermore, we show that the property $\mathrm{KLX}\le k$ is ${\mathrm{MSO}}_2$-expressible for every fixed $k$. Combined with the tree-width bound, this result implies that determining whether a graph has KLX number at most $k$ can be achieved in linear time for any constant $k$.

13 pages, excluding the 11-page appendix and the bibliography. Accepted to MFCS 2026