Contractible Edges in k-Connected Graphs with Some Forbidden Subgraphs

In 2001, Kawarabayashi proved that for any odd integer k ≥ 3, if a k-connected graph G is \({K^{-}_{4}}\) -free, then G has a k-contractible edge. He pointed out, by a counterexample, that this result does not hold when k is even. In this paper, we have proved the following two results on the subject: (1) For any even integer k ≥ 4, if a k-connected graph G is \({K_{4}^{-}}\) -free and d _G (x) + d _G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge. (2) Let t ≥ 3, k ≥ 2t – 1 be integers. If a k-connected graph G is \({(K_{1}+(K_{2} \cup K_{1, t}))}\) -free and d _G (x) + d _G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge.     

Nonseparating Induced Cycles Consisting of Contractible Edges in k-Connected Graphs

It is proved that if G is a k-connected graph which does not contain K₄⁻, then G has an induced cycle C such that G – V(C) is (k − 2)-connected and either every edge of C is k-contractible or C is a triangle. This theorem is a generalization of some known theorems.     

Contractible Non-Edges in Triangle-Free Graphs

Thomassen has proved that every triangle-free k-connected graph contains a pair of adjacent vertices whose identification yields again a k-connected graph. We study the existence of a pair of nonadjacent vertices whose identification preserves k-connectivity. In particular, we present a reduction theorem for the class of all bipartite, k-connected graphs. Revised: January 7, 1999     

Large Cliques in -Free Graphs

Dedicated to the memory of Paul Erdős A graph is called -free if it contains no cycle of length four as an induced subgraph. We prove that if a -free graph has n vertices and at least edges then it has a complete subgraph of vertices, where depends only on . We also give estimates on and show that a similar result does not hold for H-free graphs––unless H is an induced subgraph of . The best value of is determined for chordal graphs. Received October 25, 1999 RID="*" ID="*" Supported by OTKA grant T029074. RID="**" ID="**" Supported by TKI grant stochastics@TUB and by OTKA grant T026203.     

Contractible Edges in 7-Connected Graphs

An edge of a k-connected graph is said to be a k-contractible edge, if its contraction yields again a k-connected graph. A noncomplete k-connected graph possessing no k-contractible edges is called contraction critical k-connected. Recently, Kriesell proved that every contraction critical 7-connected graph has two distinct vertices of degree 7. And he guessed that there are two vertices of degree 7 at distance one or two. In this paper, we give a proof to his conjecture. The work partially supported by NNSF of China(Grant number: 10171022)     

Contractible Small Subgraphs in k-connected Graphs

We prove that if G is highly connected, then either G contains a non-separating connected subgraph of order three or else G contains a small obstruction for the above conclusion. More precisely, we prove that if G is k-connected (with k ≥ 2), then G contains either a connected subgraph of order three whose contraction results in a k-connected graph (i.e., keeps the connectivity) or a subdivision of ${K_4^-}$ whose order is at most 6.     

Contractible Triples in Highly Connected Graphs

We prove that if G is k-connected (with k ≥ 2), then G contains either a cycle of length 4 or a connected subgraph of order 3 whose contraction results in a k-connected graph. This immediately implies that any k-connected graph has either a cycle of length 4 or a connected subgraph of order 3 whose deletion results in a (k − 1)-connected graph.     

6连通图中的可收缩边

Kriesell(2001年)猜想：如果κ连通图中任意两个相邻顶点的度的和至少是2[5κ/4]-1则图中有κ-可收缩边．本文证明每一个收缩临界6连通图中有两个相邻的度为6的顶点，由此推出该猜想对κ=6成立。     

Finding All Maximal Cliques in Dynamic Graphs

Clustering applications dealing with perception based or biased data lead to models with non-disjunct clusters. There, objects to be clustered are allowed to belong to several clusters at the same time which results in a fuzzy clustering. It can be shown that this is equivalent to searching all maximal cliques in dynamic graphs like G _t = (V,E _t), where E _{t – 1} E _t, t = 1,...,T; E ₀ = . In this article algorithms are provided to track all maximal cliques in a fully dynamic graph.     

团复形是无圈的平面图是可伸缩图

许宝刚猜想:若图G的团复形是无圈的,则G为可伸缩图．本文证明了该猜想对平面图成立,即:若G是团复形为无圈的平面图,则G为可伸缩图．     

An Extremal Problem on Contractible Edges in 3-Connected Graphs

An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. The existence of contractible edges is a very useful induction tool. Let G be a simple 3-connected graph with at least five vertices. Wu [7] proved that G has at most vertices that are not incident to contractible edges. In this paper, we characterize all simple 3-connected graphs with exactly vertices that are not incident to contractible edges. We show that all such graphs can be constructed from either a single vertex or a 3-edge-connected graph (multiple edges are allowed, but loops are not allowed) by a simple graph operation. Research partially supported by an ONR grant under grant number N00014-01-1-0917     

On the Structure of Contractible Vertex Pairs in Chordal Graphs

An edge/non-edge in a k-connected graph is contractible if its contraction does not result in a graph of lower connectivity. We focus our study on contractible edges and non-edges in chordal graphs. Firstly, we characterize contractible edges in chordal graphs using properties of tree decompositions with respect to minimal vertex separators. Secondly, we show that in every chordal graph each non-edge is contractible. We also characterize non-edges whose contraction leaves a k-connected chordal graph.     

On Small Contractible Subgraphs in 3-connected Graphs of Small Average Degree

We study the distribution of small contractible subgraphs in 3-connected graphs under local regularity conditions.     

A Survey on Contractible Edges in Graphs of a Prescribed Vertex Connectivity

 The aim of the present paper is to survey old and recent results on contractible edges in graphs of a given vertex connectivity. Received: June 5, 2001 Final version received: August 23, 2001