30 research outputs found
Connectivity and tree structure in finite graphs
Considering systems of separations in a graph that separate every pair of a
given set of vertex sets that are themselves not separated by these
separations, we determine conditions under which such a separation system
contains a nested subsystem that still separates those sets and is invariant
under the automorphisms of the graph.
As an application, we show that the -blocks -- the maximal vertex sets
that cannot be separated by at most vertices -- of a graph live in
distinct parts of a suitable tree-decomposition of of adhesion at most ,
whose decomposition tree is invariant under the automorphisms of . This
extends recent work of Dunwoody and Kr\"on and, like theirs, generalizes a
similar theorem of Tutte for .
Under mild additional assumptions, which are necessary, our decompositions
can be combined into one overall tree-decomposition that distinguishes, for all
simultaneously, all the -blocks of a finite graph.Comment: 31 page
Unifying duality theorems for width parameters in graphs and matroids. II. General duality
We prove a general duality theorem for tangle-like dense objects in
combinatorial structures such as graphs and matroids. This paper continues, and
assumes familiarity with, the theory developed in [6]Comment: 19 page
-Blocks: a connectivity invariant for graphs
A -block in a graph is a maximal set of at least vertices no two
of which can be separated in by fewer than other vertices. The block
number of is the largest integer such that has a
-block.
We investigate how interacts with density invariants of graphs, such
as their minimum or average degree. We further present algorithms that decide
whether a graph has a -block, or which find all its -blocks.
The connectivity invariant has a dual width invariant, the
block-width of . Our algorithms imply the duality theorem
: a graph has a block-decomposition of width and adhesion if and only if it contains no -block.Comment: 22 pages, 5 figures. This is an extended version the journal article,
which has by now appeared. The version here contains an improved version of
Theorem 5.3 (which is now best possible) and an additional section with
examples at the en
Canonical tree-decompositions of finite graphs I. Existence and algorithms
We construct tree-decompositions of graphs that distinguish all their
k-blocks and tangles of order k, for any fixed integer k. We describe a family
of algorithms to construct such decompositions, seeking to maximize their
diversity subject to the requirement that they commute with graph isomorphisms.
In particular, all the decompositions constructed are invariant under the
automorphisms of the graph.Comment: 23 pages, 5 figure
Entanglements
Robertson and Seymour constructed for every graph a tree-decomposition
that efficiently distinguishes all the tangles in . While all previous
constructions of these decompositions are either iterative in nature or not
canonical, we give an explicit one-step construction that is canonical. The key
ingredient is an axiomatisation of 'local properties' of tangles.
Generalisations to locally finite graphs and matroids are also discussed.Comment: 8 pages, 4 figure