2026/06/23 by Codaline Bourotte, Gwendal Ducloz, Pekka Orponen +1 · 1 voice
Computer Science · #cs.DM
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 TW ≤ KLX+1. Furthermore, we show that the property KLX ≤ k is MSO2-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.