2026-09-08T02:22:02Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:26518 2026-09-05T19:46:14Z ddc:004 open_access

Thin Trees for near Minimum Cuts Klein, Nathan Olver, Neil Yeoh, Zi Song Graph Theory Thin Trees Polygon Representation Near Minimum Cuts The strong thin tree conjecture states that every k-edge-connected graph G contains an O(1/k)-thin spanning tree, meaning a spanning tree which contains at most an O(1/k) fraction of the edges across each cut in G. This conjecture is still open despite significant effort; the best current result by Anari and Oveis Gharan shows the existence of an O(polylog log n/k)-thin tree. In this work, we demonstrate that the conjecture is true if one only requires thinness for the set of η-near minimum cuts of the graph for η = 1/40, in other words, for the set of cuts with fewer than (1+1/40)k edges. Our approach constructs such a tree in polynomial time. To show this, we utilize the structure of near minimum cuts, and in particular the polygon representation of Benczúr and Goemans, to reduce to the previously solved problem of finding a spanning tree that is O(1/k)-thin for all sets in a laminar family. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Nathan Klein and Neil Olver and Zi Song Yeoh 2026 Is Part Of LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2026.129 urn:nbn:de:0030-drops-265180 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.129 eng https://creativecommons.org/licenses/by/4.0/legalcode