Hadwiger’s conjecture asserts that if a simple graph G has no Kt+1 minor, then its vertex set V (G) can be partitioned into t stable sets. This is still open, but we prove under the same hypotheses that V (G) can be partitioned into t sets X1, . . . ,Xt, such that for 1 ≤ i ≤ t, the subgraph induced on Xi has maximum degree at most a function of t. This is… (More)

In 2013 Belmonte and Vatshelle used mim-width, a graph parameter bounded on interval graphs and permutation graphs that strictly generalizes clique-width, to explain existing algorithms for many domination-type problems, also known as (σ, ρ)problems or LC-VSVP problems, on many special graph classes. In this paper, we focus on chordal graphs and… (More)

An r-matching in a graph G is a collection of edges in G such that the distance between any two edges is at least r. This generalizes both matchings and induced matchings as matchings are 1-matchings and induced matchings are 2-matchings. In this paper, we estimate the minimum and maximum number of r-matchings in a tree with fixed order.

We introduce a new graph width parameter, called special induced matching width, shortly sim-width, which does not increase when taking induced minors. For a vertex partition (A,B) of a graphG, this parameter is based on the maximum size of an induced matching {a1b1, . . . , ambm} in G where a1, . . . , am ∈ A and b1, . . . , bm ∈ B. Classes of graphs of… (More)