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13
Contractible Cliques in kConnected Graphs
, 2005
"... Kawarabayashi proved that for any integer k ≥ 4, every kconnected graph contains two triangles sharing an edge, or admits a kcontractible edge, or admits a kcontractible triangle. This implies Thomassen’s result that every trianglefree kconnected graph contains a kcontractible edge. In this pa ..."
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. In this paper, we extend Kawarabayashi’s technique and prove a more general result concerning kcontractible cliques.
Contractible Subgraphs in kConnected Graphs
, 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 kconnected graph and T(G) contains no edge then G admits a kcontractible 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 kconnected graph and T(G) contains no edge then G admits a kcontractible clique of size at most
Clique Minors In Graphs And Their Complements
, 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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Cited by 2 (0 self)
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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
 J. Graph Theory
"... Abstract. A ktree is a chordal graph with no (k + 2)clique. An ℓtreepartition of a graph G is a vertex partition of G into ‘bags’, such that contracting each bag to a single vertex gives an ℓtree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ℓ ≥ ..."
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Cited by 2 (2 self)
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Abstract. A ktree is a chordal graph with no (k + 2)clique. An ℓtreepartition of a graph G is a vertex partition of G into ‘bags’, such that contracting each bag to a single vertex gives an ℓtree (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
 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 ..."
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Cited by 17 (3 self)
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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
 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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Cited by 10 (1 self)
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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
"... Abstract. For a given graph property Π (i.e., a collection Π of graphs), the ΠContraction 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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Cited by 2 (1 self)
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Abstract. For a given graph property Π (i.e., a collection Π of graphs), the ΠContraction 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
, 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 FixedParameter Tractable with respect to k ..."
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Cited by 1 (1 self)
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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
"... 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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Cited by 1 (0 self)
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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 rightangled Artin groups
, 2011
"... Abstract. In this article we study the rightangled 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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Cited by 12 (3 self)
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Abstract. In this article we study the rightangled 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
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