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Results 1 - 10
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13
Contractible Cliques in k-Connected Graphs
, 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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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
Contractible Subgraphs in k-Connected 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 k-connected graph and T(G) contains no edge then G admits a k-contractible clique of size at most ..."
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3, generalizing an earlier result of Thomassen. In this paper, we further generalize Kawarabayashi’s result by showing that if G is k-connected and the maximum degree of T(G) is at most 1, then G admits a k-contractible clique of size at most 3 or there exist independent edges e and f of G
Planarity allowing few error vertices in linear time
"... Abstract — We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has a vertex set X of order at most k such that G − X is planar (we call this class of graphs k-apex), and if this is the case, computes a drawing of the graph in the plane after del ..."
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Cited by 9 (1 self)
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the question posed by Cabello and Mohar in 2005, and by Kawarabayashi and Reed (STOC’07), respectively. Note that the case k = 0 is the planarity case. Thus our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan (J. ACM 1974), which determines if a given graph is planar
A.A.Osipov †
, 2008
"... A model of the ρ-meson as a collective excitation of q¯q pairs in a system that obeys the modified Nambu–Jona-Lasinio Lagrangian is proposed. The ρ emerges as a dormant Goldstone boson. The origin of the ρ-meson mass is understood as a result of spontaneous chiral symmetry breaking. The low-energy d ..."
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A model of the ρ-meson as a collective excitation of q¯q pairs in a system that obeys the modified Nambu–Jona-Lasinio Lagrangian is proposed. The ρ emerges as a dormant Goldstone boson. The origin of the ρ-meson mass is understood as a result of spontaneous chiral symmetry breaking. The low
Complete Minors, Independent Sets, and Chordal Graphs
, 2009
"... The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger’s Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that α(G)h(G) ≥ |V (G)|. We show that (2α(G) − ⌈log τ (τα(G)/2)⌉)h(G) ≥ |V (G) | where τ ≈ 6.83. For graphs w ..."
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Cited by 6 (1 self)
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with α(G) ≥ 14, this improves on a recent result of Kawarabayashi and Song who showed (2α(G) − 2)h(G) ≥ |V (G) | when α(G) ≥ 3.
Algorithmic Meta-Theorems
- In M. Grohe and R. Neidermeier eds, International Workshop on Parameterized and Exact Computation (IWPEC), volume 5018 of LNCS
, 2008
"... Algorithmic meta-theorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of ..."
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Cited by 22 (6 self)
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Algorithmic meta-theorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C
CONTRACTIBILITY AND THE HADWIGER CONJECTURE
, 2008
"... Consider the following relaxation of the Hadwiger Conjecture: For each t there exists Nt such that every graph with no Kt-minor admits a vertex partition into ⌈αt+β ⌉ parts, such that each component of the subgraph induced by each part has at most Nt vertices. The Hadwiger Conjecture corresponds to ..."
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Cited by 3 (2 self)
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to Kawarabayashi and Mohar, (2) a recent result of Norine and Thomas that says that every sufficiently large (t + 1)-connected graph contains a Kt-minor, and (3) a new sufficient condition for a graph to have a set of edges whose contraction increases the connectivity.
Polynomial Bounds for the Grid-Minor Theorem
, 2013
"... One of the key results in Robertson and Seymour’s seminal work on graph minors is the Grid-Minor Theorem. The theorem states that any graph of treewidth at least k contains a k ′ ×k ′ grid as a minor where k ′ ≥ f(k) for some function f. This theorem has found many applications in graph theory and ..."
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Cited by 15 (3 self)
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One of the key results in Robertson and Seymour’s seminal work on graph minors is the Grid-Minor Theorem. The theorem states that any graph of treewidth at least k contains a k ′ ×k ′ grid as a minor where k ′ ≥ f(k) for some function f. This theorem has found many applications in graph theory
A Simple Algorithm for the Graph Minor Decomposition -- Logic meets Structural Graph Theory
"... A key result of Robertson and Seymour’s graph minor theory is a structure theorem stating that all graphs excluding some fixed graph as a minor have a tree decomposition into pieces that are almost embeddable in a fixed surface. Most algorithmic applications of graph minor theory rely on an algorith ..."
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Cited by 7 (2 self)
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A key result of Robertson and Seymour’s graph minor theory is a structure theorem stating that all graphs excluding some fixed graph as a minor have a tree decomposition into pieces that are almost embeddable in a fixed surface. Most algorithmic applications of graph minor theory rely
Fractional coloring and the odd Hadwiger’s conjecture
, 2008
"... Gerards and Seymour (see [T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley-Interscience, 1995], page 115) conjectured that if a graph has no odd complete minor of order p, then it is (p − 1)-colorable. This is an analogue of the well known conjecture of Hadwiger, and in fact, this would immediat ..."
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Cited by 2 (0 self)
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immediately imply Hadwiger’s conjecture. The current best known bound for the chromatic number of graphs without an odd complete minor of order p is O(p √ log p) by the recent result by Geelen et al. [J. Geelen, B. Gerards, B. Reed, P. Seymour, A. Vetta, On the odd variant of Hadwiger’s conjecture (submitted
Results 1 - 10
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13
