Results 1 - 10 of 13

Contractible Cliques in k-Connected Graphs

by Xiaolong Huang, Zemin Jin, Xingxing Yu, Xiaoyan Zhang , 2005
"... Kawarabayashi proved that for any integer k ≥ 4, every k-connected graph contains two triangles sharing an edge, or admits a k-contractible edge, or admits a k-contractible triangle. This implies Thomassen’s result that every triangle-free k-connected graph contains a k-contractible edge. In this pa ..."
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. In this paper, we extend Kawarabayashi’s technique and prove a more general result concerning k-contractible cliques.

Contractible Subgraphs in k-Connected Graphs

by Zemin Jin, Xingxing Yu, Xiaoyan Zhang , 2006
"... For a graph G we define a graph T(G) whose vertices are the triangles in G and two vertices of T(G) are adjacent if their corresponding triangles in G share an edge. Kawarabayashi showed that if G is a k-connected graph and T(G) contains no edge then G admits a k-contractible clique of size at most ..."
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For a graph G we define a graph T(G) whose vertices are the triangles in G and two vertices of T(G) are adjacent if their corresponding triangles in G share an edge. Kawarabayashi showed that if G is a k-connected graph and T(G) contains no edge then G admits a k-contractible clique of size at most

Clique Minors In Graphs And Their Complements

by Bruce Reed, Robin Thomas , 2000
"... A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let t ≥ 1 be an integer, and let G be a graph on n vertices with no minor isomorphic to Kt+1. Kostochka conjectures that there exists a constant c = c(k) independent of G such that the complement of G h ..."
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A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let t ≥ 1 be an integer, and let G be a graph on n vertices with no minor isomorphic to Kt+1. Kostochka conjectures that there exists a constant c = c(k) independent of G such that the complement of G

Vertex partitions of chordal graphs

by David R. Wood - J. Graph Theory
"... Abstract. A k-tree is a chordal graph with no (k + 2)-clique. An ℓ-treepartition of a graph G is a vertex partition of G into ‘bags’, such that contracting each bag to a single vertex gives an ℓ-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ℓ ≥ ..."
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Abstract. A k-tree is a chordal graph with no (k + 2)-clique. An ℓ-treepartition of a graph G is a vertex partition of G into ‘bags’, such that contracting each bag to a single vertex gives an ℓ-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ℓ

EFFICIENT DATA STRUCTURE FOR REPRESENTING AND SIMPLIFYING SIMPLICIAL COMPLEXES IN HIGH DIMENSIONS

by Dominique Attali, André Lieutier, David Salinas - INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS , 2012
"... We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the link condition can be contracted while preserving the homotopy type. Our primary interest is to simplify flag complexes such a ..."
Abstract - Cited by 17 (3 self) - Add to MetaCart
edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex K by the graph G of its edges

Potential of the Approximation Method

by Kazuyuki Amano, Akira Maruoka - Proc. of the 37th IEEE Symp. on the Foundations of Computer Science , 1996
"... Developing some techniques for the approximation method, we establish precise versions of the following statements concerning lower bounds for circuits that detect cliques of size s in a graph with m vertices: For 5 s m=4, a monotone circuit computing CLIQUE(m; s) contains at least (1=2)1:8 min( ..."
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the latter is verified introducing a notion of restricting negation and generalizing the sunflower contraction. 1. Introduction Since Razborov, based on the approximation method, succeeded to obtain a superpolynomial lower bound on the size of monotone circuits computing the clique function, much effort

Contracting Few Edges to Remove Forbidden Induced Subgraphs

by Leizhen Cai, Chengwei Guo
"... Abstract. For a given graph property Π (i.e., a collection Π of graphs), the Π-Contraction problem is to determine whether the input graph G can be transformed into a graph satisfying property Π by contracting at most k edges, where k is a parameter. In this paper, we mainly focus on the parameteriz ..."
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Abstract. For a given graph property Π (i.e., a collection Π of graphs), the Π-Contraction problem is to determine whether the input graph G can be transformed into a graph satisfying property Π by contracting at most k edges, where k is a parameter. In this paper, we mainly focus

Treewidth reduction for the parameterized Multicut problem

by Jean Daligault, Christophe Paul, Anthony Perez, Stéphan Thomassé , 2010
"... The parameterized Multicut problem consists in deciding, given a graph, a set of requests (i.e. pairs of vertices) and an integer k, whether there exists a set of k edges which disconnects the two endpoints of each request. Determining whether Multicut is Fixed-Parameter Tractable with respect to k ..."
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an irrelevant request that can be safely removed. As a main consequence, these rules imply that the degree of the request graph of any instance is bounded by a function of k. We prove that when the input graph has a large clique minor or a large grid minor, then we can remove an irrelevant request or contract

ON THE TOPOLOGY OF WEAKLY AND STRONGLY SEPARATED SET COMPLEXES

by Daniel Hess, Benjamin Hirsch
"... We examine the topology of the clique complexes of the graphs of weakly and strongly separated subsets of the set [n] = {1, 2,..., n}, which, after deleting all cone points, we denote by ˆ ∆ws(n) and ˆ ∆ss(n), respectively. In particular, we find that ˆ ∆ws(n) is contractible for n ≥ 4, while ˆ ∆s ..."
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We examine the topology of the clique complexes of the graphs of weakly and strongly separated subsets of the set [n] = {1, 2,..., n}, which, after deleting all cone points, we denote by ˆ ∆ws(n) and ˆ ∆ss(n), respectively. In particular, we find that ˆ ∆ws(n) is contractible for n ≥ 4, while ˆ

Embedability between right-angled Artin groups

by Sang-hyun Kim, Thomas Koberda , 2011
"... Abstract. In this article we study the right-angled Artin subgroups of a given rightangled Artin group. Starting with a graph Γ, we produce a new graph through a purely combinatorial procedure, and call it the extension graph Γ e of Γ. We produce a second graph Γ e k, the clique graph of Γe, by addi ..."
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Abstract. In this article we study the right-angled Artin subgroups of a given rightangled Artin group. Starting with a graph Γ, we produce a new graph through a purely combinatorial procedure, and call it the extension graph Γ e of Γ. We produce a second graph Γ e k, the clique graph of Γe
Results 1 - 10 of 13