Tree Containment above Minimum Degree Is FPT

Open Webpage

Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, Kirill Simonov

Published: 2026, Last Modified: 06 May 2026ACM Trans. Algorithms 2026EveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: According to the classic Chvátal’s Lemma from 1977, a graph \(G\) of minimum degree \(\delta(G)\) contains every tree on \(\delta(G)+1\) vertices. Our main result is the following algorithmic “extension” of Chvátal’s Lemma: For any \(n\)-vertex graph \(G\), an integer \(k\), and a tree \(T\) on at most \(\delta(G)+k\) vertices, deciding whether \(G\) contains a subgraph isomorphic to \(T\) can be done in time \(f(k)\cdot n^{\mathcal{O}(1)}\) for some function \(f\) of \(k\) only. The proof is based on an intricate interplay between extremal graph theory and parameterized algorithms.