KAIST Discrete Math Seminar

We prove that if T₁,…, T_n is a sequence of bounded degree trees so that T_i has i vertices, then K_n has a decomposition into T₁,…, T_n. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees.
We deduce this result from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs.
In this talk, we discuss the ideas used in the proof.
This is joint work with Felix Joos, Daniela Kühn and Deryk Osthus.

It has long been known that Feedback Vertex Set can be solved in time 2^{O(w log w)}n^O(1) on graphs of treewidth w, but it was only recently that this running time was improved to 2^O(w)n^O(1), that is, to single-exponential parameterized by treewidth. We consider a natural generalization of this problem, which asks given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices of G such that each block of G-S has at most d vertices. The central question of this talk is: “Can we obtain an algorithm that runs in single-exponential time parameterized by treewidth, for every fixed d?” The answer is negative. But then, one may be curious which properties of Feedback Vertex Set that make it allow to have a single-exponential algorithm. To answer this question, we add an additional condition in the general problem, and provide a dichotomy result.
Formally, for a class ? of graphs, the Bounded ?-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices of G such that each block of G-S has at most d vertices and is in ?. Assuming that ? is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d:

• if ? consists only of chordal graphs, then the problem can be solved in time 2^{O(wd^2)}n^O(1),
• if ? contains a graph with an induced cycle of length ℓ≥4, then the problem is not solvable in time 2^{o(w log w)} n^O(1) even for fixed d=ℓ, unless the ETH fails.

We further show that it is W[1]-hard parameterized by only treewidth, when ? consists of all graphs. This is joint work with Édouard Bonnet, Nick Brettell, and Daniel Marx.