A Congestion Parameter for Depth-First Graph Traversals

2026-06-23

AI Summary

This study addresses the problem of minimizing edge congestion in depth-first search (DFS) traversals, motivated by applications in RNA nanostructure design and its intrinsic theoretical interest. The authors introduce the KLX number—a novel graph invariant defined as the maximum number of back edges simultaneously active during a DFS traversal—and establish key structural properties: the treewidth TW satisfies TW ≤ KLX + 1, and the property KLX ≤ k is expressible in monadic second-order logic (MSO₂). Leveraging graph-theoretic analysis and formal methods, the paper provides complete characterizations for graph classes with KLX numbers 0, 1, and 2, along with linear-time recognition algorithms. Furthermore, it proves that for any fixed k, deciding whether KLX ≤ k is solvable in linear time.

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 $k$.

Problem

• graph traversal
• edge congestion
• depth-first search
• KLX number
• graph parameter

Innovation

• KLX number
• depth-first search
• edge congestion
• tree-width
• MSO₂-expressibility