Tree Containment above Minimum Degree Is FPT

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.