Vertex Cuts

Given a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings' theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2‐connected graphs, that are not 3‐connected. Most of these structure tree theories have been based on edge cuts, which are components of the graph obtained by removing finitely many edges. A new axiomatic theory is described here using vertex cuts, components of the graph obtained by removing finitely many vertices. This generalizes Tutte's decomposition of 2‐connected graphs to k‐connected graphs for any k, in finite and infinite graphs. The theory can be applied to nonlocally finite graphs with more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a decomposition for a group acting on such a graph, generalizing Stallings' theorem. Further applications include the classification of distance transitive graphs and k‐CS‐transitive graphs.     

Super-connected edge transitive graphs

A graph G is said to be super-connected if any minimum cut of G isolates a vertex. In a previous work due to the second author of this note, super-connected graphs which are both vertex transitive and edge transitive are characterized. In this note, we generalize the characterization to edge transitive graphs which are not necessarily vertex transitive, showing that the only irreducible edge transitive graphs which are not super-connected are the cycles C_n(n?6) and the line graph of the 3-cube, where irreducible means the graph has no vertices with the same neighbor set. Furthermore, we give some sufficient conditions for reducible edge transitive graphs to be super-connected.     

A vertex incremental approach for maintaining chordality

For a chordal graph G=(V,E), we study the problem of whether a new vertex u∉V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv, a characterization of the set of edges incident to u that also must be added to G along with uv. We propose a data structure that can compute and add each such set in O(n) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is on-line in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently.     

Minimal separating sets of maximum size

A connected graph G can be disconnected or reduced to a single vertex by removing an appropriate subset of the vertex set V(G), and can be disconnected by removing a suitable subset of the edge set E(G). Attention has usually been centered on separating sets having minimum cardinality, and parameters called the vertex connectivity and the edge connectivity defined. These classical concepts are generalized by using separating sets which are minimal. By considering the maximum as well as the minimum cardinality of such sets, one defines vertex and edge connectivity parameters. Sharp upper bounds are established for these numbers and their values computed for certain classes of graphs. An analogue of Whitney's theorem on connectivity is obtained. Parameters are also defined for minimal separating sets consisting of a mixture of vertices and edges, and these are shown to depend on the maximum and minimum values of the vertex and edge connectivity parameters.     

图的半强积的邻点可区别染色

两个简单图G与H的半强积G·H是具有顶点集V(G)×V(H)的简单图,其中两个顶点(u,v)与(u',v')相邻当且仅当u=u'且vv'∈E(H),或uu'∈E(G)且vv'∈E(H).图的邻点可区别边(全)染色是指相邻点具有不同色集的正常边(全)染色.统称图的邻点可区别边染色与邻点可区别全染色为图的邻点可区别染色.图G的邻点可区别染色所需的最少的颜色数称为邻点可区别染色数,并记为X_a~((r))(G),其中r=1,2,且X_a~((1))(G)与X_a~((2))(G)分别表示G的邻点可区别的边色数与全色数.给出了两个简单图的半强积的邻点可区别染色数的一个上界,并证明了该上界是可达的.然后,讨论了两个树的不同半强积具有相同邻点可区别染色数的充分必要条件.另外,确定了一类图与完全图的半强积的邻点可区别染色数的精确值.     

On the adjacent vertex-distinguishing acyclic edge coloring of some graphs

A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by x′ _Aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.     

Minimal split completions

We study the problem of adding an inclusion minimal set of edges to a given arbitrary graph so that the resulting graph is a split graph, called a minimal split completion of the input graph. Minimal completions of arbitrary graphs into chordal and interval graphs have been studied previously, and new results have been added recently. We extend these previous results to split graphs by giving a linear-time algorithm for computing minimal split completions. We also give two characterizations of minimal split completions, which lead to a linear time algorithm for extracting a minimal split completion from any given split completion.We prove new properties of split graph that are both useful for our algorithms and interesting on their own. First, we present a new way of partitioning the vertices of a split graph uniquely into three subsets. Second, we prove that split graphs have the following property: given two split graphs on the same vertex set where one is a subgraph of the other, there is a sequence of edges that can be removed from the larger to obtain the smaller such that after each edge removal the modified graph is split.     

Circuit decompositions of join‐covered graphs

In this paper, we focus our attention on join‐covered graphs, that is, ±1‐weighted graphs, without negative circuits, in which every edge lies in a zero‐weight circuit. Join covered graphs are a natural generalization of matching‐covered graphs. Many important properties of matching covered graphs, such as the existence of a canonical partition, tight cut decomposition and ear decomposition, have been generalized to join covered graphs by A. Seb? [5]. In this paper we prove that any two edges of a join‐covered graph lie on a zero‐weight circuit (under an equivalent weighting), generalize this statement to an arbitrary number of edges, and characterize minimal bipartite join‐covered graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 220–233, 2009     

A Vizing-type theorem for matching forests

A well-known Theorem of Vizing states that one can colour the edges of a graph by Δ+ colours, such that edges of the same colour form a matching. Here, Δ denotes the maximum degree of a vertex, and the maximum multiplicity of an edge in the graph. An analogue of this Theorem for directed graphs was proved by Frank. It states that one can colour the arcs of a digraph by Δ+ colours, such that arcs of the same colour form a branching. For a digraph, Δ denotes the maximum indegree of a vertex, and the maximum multiplicity of an arc. We prove a common generalization of the above two theorems concerning the colouring of mixed graphs (these are graphs having both directed and undirected edges) in such a way that edges of the same colour form a matching forest.     

Forbidden minors for wye‐delta‐wye reducibility

A graph is Y Δ Y reducible if it can be reduced to a single vertex by a sequence of series‐parallel reductions and Y Δ Y transformations. The class of Y Δ Y reducible graphs is minor closed. We found a large number of minor minimal Y Δ Y irreducible graphs: a family of 57578 31‐edge graphs and another 40‐edge graph. It is still an open problem to characterize Y Δ Y reducible graphs in terms of a finite set of forbidden minors. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 317–321, 2004     

Vertex 2-isomorphism

Whitney's theorem on 2-isomorphism characterizes the set of graphs having the same cycles as a given graph, where a cycle is regarded as a set of edges. In this paper, vertex 2-isomorphism is defined and used to prove a vertex analogue of Whitney's theorem. The main theorem states that two connected graphs have the same set of cycles, where a cycle is now regarded as a set of vertices, if and only if one can be obtained from the other by a sequence of simple operations. © 1992 John Wiley & Sons, Inc.     

关于图的点可区别边染色猜想的一点注

图G的一个k-正常边染色f被称为点可区别的是指任意两点的点及其关联边所染色集合不同,所用最少颜色数被称为G的点可区别边色数,张忠辅教授提出一个猜想即对每一个正整数k≥3,总存在一个最大度为△(G)=k≥3的图G,图G一定有一个子图H,使得G的点可区别的边色数不超过子图的.本文证明了对于最大度△≤6时,猜想正确.     

A Jump Theorem with Uniform Estimates for \documentclass{article}\pagestyle{empty}\begin{document}$\mathop \partial \limits_b^ - $\end{document}b-closed Forms on Real Hypersurfaces

Define the partial join of two graphs to be some graph arising from their disjoint union by adding a set of new edges each joining a vertex of the first graph and a vertex of the second one. We characterize all colour-critical graphs being partial joins of a complete graph and an odd cycle thus completely answering a special case of a question raised by T. GALLAI in 1969.     

A characterization of well-covered graphs that contain neither 4- nor 5-cycles

A graph is well covered if every maximal independent set has the same cardinality. A vertex x, in a well-covered graph G, is called extendable if G – {x} is well covered and β(G) = β(G – {x}). If G is a connected, well-covered graph containing no 4- nor 5-cycles as subgraphs and G contains an extendable vertex, then G is the disjoint union of edges and triangles together with a restricted set of edges joining extendable vertices. There are only 3 other connected, well-covered graphs of this type that do not contain an extendable vertex. Moreover, all these graphs can be recognized in polynomial time.     

The incidence matrix and labelings of a graph

We deal with the problem of labeling the edges of a graph in such a way that the labels of the edges incident with any vertex add up to a value prescribed for that vertex. We show that the use of elementary column operations on the incidence matrix is fruitful in giving easy proofs of theorems on magic graphs and labeling |1, 3, 4|. The method can be visualized in the graph and also leads to a simple proof of a theorem in |2| on the multiplicity of −2 as an eigenvalue of a line graph. We also deal with mixed graphs, the label of a directed edge being subtracted at its initial vertex.     

Augmenting the Edge‐Connectivity of a Hypergraph by Adding a Multipartite Graph

Given a hypergraph, a partition of its vertex set, and a nonnegative integer k, find a minimum number of graph edges to be added between different members of the partition in order to make the hypergraph k‐edge‐connected. This problem is a common generalization of the following two problems: edge‐connectivity augmentation of graphs with partition constraints (J. Bang‐Jensen, H. Gabow, T. Jordán, Z. Szigeti, SIAM J Discrete Math 12(2) (1999), 160–207) and edge‐connectivity augmentation of hypergraphs by adding graph edges (J. Bang‐Jensen, B. Jackson, Math Program 84(3) (1999), 467–481). We give a min–max theorem for this problem, which implies the corresponding results on the above‐mentioned problems, and our proof yields a polynomial algorithm to find the desired set of edges.     

某些中间图的邻点可区别E-全色数

设G（V,E）是简单连通图,T（G）为图G的所有顶点和边构成的集合,并设C是k-色集（k是正整数）,若T（G）到C的映射f满足：对任意uv∈E（G）,有f（u）≠f（v）,f（u）≠f（uv）,f（v）≠f（uv）,并且C（u）≠C（v）,其中C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.那么称f为图G的邻点可区别E-全染色（简记为k-AVDETC）,并称χ_（at）~e（G）=min{k｜图G有k-邻点可区别E-全染色}为G的邻点可区别E-全色数.图G的中间图M（G）就是在G的每一个边上插入一个新的顶点,再把G上相邻边上的新的顶点相联得到的.探讨了路、圈、扇、星及轮的中间图的邻点可区别E-全染色,并给出了这些中间图的邻点可区别E-全色数.     

Minimal triangulations of graphs: A survey

Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.     

关于积图点可区别边染色的若干结论

如果图G的一个正常边染色满足任意两个不同点的关联边色集不同, 则称为点可区别边染色(VDEC), 其所用最少颜色数称为点可区别边色数. 利用构造法给出了积图点可区别边染色的一个结论, 得到了关于积图点可区别边色数的若干结果, 并且给出25个具体积图的点可区别边色数, 验证了它们满足点可区别边染色猜想(VDECC).     

图与超图中的彩色匹配综述

超图H=(V,E)是一个二元组(V,E),其中超边集E中的元素是点集V的非空子集.因此图是一种特殊的超图,超图也可以看作是一般图的推广.特别地,如果超边集E中的元素均是点集V的k元子集,则称该超图为k-一致的.通常情况下,为叙述简便,我们也会将超边简称为边.图(超图)中的匹配是指图(超图)中互不相交的边的集合.对于图(超图)中的彩色匹配,有两种定义方式:一为染色图(超图)中互不相交且颜色不同的边的集合;二为顶点集均为[n]的多个染色图(超图)所构成的集族中互不相交且颜色均不同的边的集合,且每条边均来自集族中不同的图(超图).现主要介绍了图与超图中关于彩色匹配的相关结果.