k-Blocks: a connectivity invariant for graphs

Johannes Carmesin; Reinhard Diestel; Matthias Hamann; Fabian Hundertmark

doi:10.1137/130923646

k-Blocks: a connectivity invariant for graphs

Johannes Carmesin, Reinhard Diestel, Matthias Hamann, Fabian Hundertmark

Mathematics

Research output: Contribution to journal › Article › peer-review

13 Citations (Scopus)

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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 fewer than k other vertices. The block number β(G) of g is the largest integer k such that g has a k-block. We investigate how β interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a k-block, or which find all its k-blocks. The connectivity invariant β(G) has a dual width invariant, the block-width bw(G) of G. Our algorithms imply the duality theorem β = bw: a graph has a block-decomposition of width and adhesion <k if and only if it contains no k-block.

Original language	English
Pages (from-to)	1876-1891
Number of pages	16
Journal	SIAM Journal on Discrete Mathematics
Volume	28
Issue number	4
Early online date	23 Oct 2014
DOIs	https://doi.org/10.1137/130923646
Publication status	Published - 2014

Bibliographical note

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 end

Keywords

math.CO
graph
connectivity
global
k-block
tangle
tree-decomposition
width parameter
tree-width

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Cite this

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title = "k-Blocks: a connectivity invariant for 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 fewer than k other vertices. The block number β(G) of g is the largest integer k such that g has a k-block. We investigate how β interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a k-block, or which find all its k-blocks. The connectivity invariant β(G) has a dual width invariant, the block-width bw(G) of G. Our algorithms imply the duality theorem β = bw: a graph has a block-decomposition of width and adhesion <k if and only if it contains no k-block. ",

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AU - Carmesin, Johannes

AU - Diestel, Reinhard

AU - Hamann, Matthias

AU - Hundertmark, Fabian

N1 - 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 end

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N2 - 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 fewer than k other vertices. The block number β(G) of g is the largest integer k such that g has a k-block. We investigate how β interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a k-block, or which find all its k-blocks. The connectivity invariant β(G) has a dual width invariant, the block-width bw(G) of G. Our algorithms imply the duality theorem β = bw: a graph has a block-decomposition of width and adhesion <k if and only if it contains no k-block.

AB - 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 fewer than k other vertices. The block number β(G) of g is the largest integer k such that g has a k-block. We investigate how β interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a k-block, or which find all its k-blocks. The connectivity invariant β(G) has a dual width invariant, the block-width bw(G) of G. Our algorithms imply the duality theorem β = bw: a graph has a block-decomposition of width and adhesion <k if and only if it contains no k-block.

KW - math.CO

KW - graph

KW - connectivity

KW - global

KW - k-block

KW - tangle

KW - tree-decomposition

KW - width parameter

KW - tree-width

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EP - 1891

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 4

ER -

k-Blocks: a connectivity invariant for graphs

Abstract

Bibliographical note

Keywords

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