KLX is the min-max congestion of open back edges over DFS traversals; graphs with KLX at most 2 are fully characterized with linear-time recognition, any graph has tree-width at most KLX+1, and KLX ≤ k is MSO2-expressible hence linear-time decidable for fixed k.
and Meunier, Pierre-Etienne and Schabanel, Nicolas and Seki, Shinnosuke , editor =
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A Congestion Parameter for Depth-First Graph Traversals
KLX is the min-max congestion of open back edges over DFS traversals; graphs with KLX at most 2 are fully characterized with linear-time recognition, any graph has tree-width at most KLX+1, and KLX ≤ k is MSO2-expressible hence linear-time decidable for fixed k.