On the Block Number of Graphs

Abstract

A k-block in a graph G is a maximal set of at least k vertices no two of which can be separated in G by deleting fewer than k vertices. The block number β(G) of G is the maximum integer k for which G contains a k-block.We prove a structure theorem for graphs without a (k+1)-block, showing that every such graph has a tree-decomposition in which every torso has at most k vertices of degree 2k 2 or greater. This yields a qualitative duality, since every graph that admits such a decomposition has block number at most 2k 2 .We also study k-blocks in graphs from classes of graphs G that exclude some fixed graph as a topological minor, and prove that every G ∈ G satisfies β(G) ≤ c 3 |G| for some constant c = c(G).Moreover, we show that every graph of tree-width at least 2k 2 has a minor containing a k-block. This bound is best possible up to a multiplicative constant.

In absence of long chordless cycles, large tree-width becomes a local phenomenon

“…We now show that these are not all just scattered about the graph. It was shown by the author in [19] that either there is a kblock or there is a tree-decomposition which separates the set of vertices of high degree into small pieces. This also follows, without explicit bounds, from a far more general result of Dvořák [5].…”

A Short Derivation of the Structure Theorem for Graphs with Excluded Topological Minors

“…It is also not hard to see that k-atomic tree-decompositions are tight. In [20], the second author used k-atomic tree-decompositions to prove a structure theorem for graphs without a k-block. In fact, the proof given there yields the following:…”

“…The converse is not true for any k ≥ 4, as there exist planar graphs with arbitrarily large blocks. The second author [20] proved a structure theorem for graphs without a k-block: 20]). Let G be a graph and k ≥ 2.…”

Edge degeneracy: Algorithmic and structural results

“…gives the classic invariant of graph degeneracy [5,18,20] and, when s = ∞, gives the parameter of ∞-admissibility that was introduced by Dvořák in [10] and studied in [6,9,15,17,21,22,24].…”

“…, ρ} and then remove some of the edges between the identified vertices. While the constants of Proposition 1.1 where not specified in [9], an alternative proof was recently given by Weißauer in [24] where d k = k, c k = k, and a k = 2k(k − 1).…”