### Citations

39 |
A theory of 3-connected graphs
- Tutte
(Show Context)
Citation Context ...pectively, K), denoted by G/e (respectively, G/K), is still k-connected. For two edges e, f of G, we use G/e/f to denote the graph obtained from G by first contracting e and then contracting f. Tutte =-=[10]-=- showed that K4 is the only 3-connected graph which does not admit any 3contractible edge. Fontet [4] and Martinov [8] proved independently that if a 4-connected graph contains no 4-contractible edge ... |

15 |
Non-separating cycles in k-connected graphs,
- Thomassen
- 1981
(Show Context)
Citation Context ... line graph of a cyclically 4-edge-connected cubic graph. For k ≥ 5, it is seemingly difficult to characterize those k-connected graphs that do not contain any k-contractible edge. However, Thomassen =-=[9]-=- showed that if a k-connected graph is triangle-free then it contains a k-contractible edge. In fact, Egawa, Enomoto, and Saito [3] showed that every k-connected triangle-free graph G contains at leas... |

13 |
A survey on contractible edges in graphs of a prescribed vertex connectivity
- Kriesell
(Show Context)
Citation Context ...4-connected. 2This means that 4-connected graphs can be reduced to K5 or the square of a 6-cycle by a sequence of contractions of edges, triangles, and independent edges. In fact, Kriesell showed in =-=[7]-=- that 4-connected graphs can be reduced to K5 or the square of a 6-cycle by a sequence of contractions of one edge or two edges. Since loops and multiple edges do not affect connectivity, we only cons... |

5 |
Contractible edges and triangles in k-connected graphs
- Kawarabayashi
(Show Context)
Citation Context ... proved that when k is odd then every k-connected K − 4 -free graph contains a k-contractible edge. Kawarabayashi also observed that the same result doesn’t hold when k is even. However, he proved in =-=[6]-=- that every k-connected K − 4 -free graph contains a k-contractible clique of size at most 3. Recall that a clique in a graph is a maximal complete subgraph. It appears that the existence of a k-contr... |

5 |
Uncontractable 4-connected graphs
- Martinov
- 1982
(Show Context)
Citation Context ...e the graph obtained from G by first contracting e and then contracting f. Tutte [10] showed that K4 is the only 3-connected graph which does not admit any 3contractible edge. Fontet [4] and Martinov =-=[8]-=- proved independently that if a 4-connected graph contains no 4-contractible edge then it is the square of a cycle of length at least 5 or it is the line graph of a cyclically 4-edge-connected cubic g... |

4 |
Cycles in k-connected graphs whose deletion results in a (k
- Egawa
- 1987
(Show Context)
Citation Context ...f which consists of a only. However, this is impossible since W3 ̸= ∅ ̸= H1. ✷ 4 Proof of the main result In this section, we complete the proof of Theorem (1.1), using an argument similar to that of =-=[2]-=-. Let k ≥ 5 be an integer and let G be a k-connected graph such that ∆(T(G)) ≤ 1. If Ce(G) = ∅ for some lonely edge e, or CT(G) = ∅ for some isolated triangle T, or Ce,f(G) = ∅ for some cohesive edges... |

3 |
Note on k-contractible edges in k-connected graphs, Australas
- Kawarabayashi
(Show Context)
Citation Context ...ry k-connected triangle-free graph G contains at least min{|V (G)| + 3 2k2 − 3k, |E(G)|} k-contractible edges. Let K − 4 denote the graph obtained from K4 by deleting an edge. Recently, Kawarabayashi =-=[5]-=- proved that when k is odd then every k-connected K − 4 -free graph contains a k-contractible edge. Kawarabayashi also observed that the same result doesn’t hold when k is even. However, he proved in ... |

2 |
Contractible edges in triangle-free graphs, Combinatorica 6
- Egawa, Enomoto, et al.
- 1986
(Show Context)
Citation Context ...aphs that do not contain any k-contractible edge. However, Thomassen [9] showed that if a k-connected graph is triangle-free then it contains a k-contractible edge. In fact, Egawa, Enomoto, and Saito =-=[3]-=- showed that every k-connected triangle-free graph G contains at least min{|V (G)| + 3 2k2 − 3k, |E(G)|} k-contractible edges. Let K − 4 denote the graph obtained from K4 by deleting an edge. Recently... |

1 |
Some forbidden subgraph conditions for a graph to have a k-contractible edge, Discrete Math
- Ando, Kawarabayashi
- 2003
(Show Context)
Citation Context ...a graph is a maximal complete subgraph. It appears that the existence of a k-contractible subgraph is related to the number of triangles containing a common edge. Also several results are obtained in =-=[1]-=- concerning the existence of contractible edges by forbidding certain subgraphs involving triangles. Therefore, for a given graph G, we define a new graph T(G) whose vertices are triangles in G, and t... |