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.     

Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs

Given a vertex r of a 3-connected graph G, we show how to find three independent spanning trees of G rooted at r. Our proof is based on showing that every 3-connected graph has a nonseparating ear decomposition. This extends Whitney's characterisation that a graph is 2-connected iff it has an ear decomposition. We also show that a nonseparating ear decomposition can be constructed in O(VE) time, and hence, three independent spanning trees can be found in O(VE) time. We construct a nonseparating ear decomposition by solving the following problem at most V times. Given an edge tr and a vertex u of a 3-connected graph G, find a nonseparating induced cycle of G through tr and avoiding u. W. T. Tutte (Proc. London Math. Soc. 13 (1963), 743–767) first showed that such a cycle can always be found. We give a linear time algorithm for this.     

Planarity and duality of finite and infinite graphs

We present a short proof of the following theorems simultaneously: Kuratowski's theorem, Fary's theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski's theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitney's theorem and MacLane's theorem this has already been done by Tutte). In connection with Tutte's planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte's condition in terms of overlap graphs is equivalent to Kuratowski's condition, we characterize completely the infinite graphs satisfying MacLane's condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte's criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.     

Heavy fans, cycles and paths in weighted graphs of large connectivity

A set of paths joining a vertex y and a vertex set L is called (y,L)-fan if any two of the paths have only y in common, and its width is the number of paths forming it. In weighted graphs, it is known that the existence of heavy fan is useful to find a heavy cycle containing some specified vertices.In this paper, we show the existence of heavy fans with large width containing some specified vertices in weighted graphs of large connectivity, which is a weighted analogue of Perfect's theorem. Using this, in 3-connected weighted graphs, we can find heavy cycles containing three specified vertices, and also heavy paths joining two specified vertices containing two more specified vertices. These results extend the previous results in 2-connected weighted graphs to 3-connected weighted graphs.     

On matroid connectivity

Three types of matroid connectivity, including Tutte's, are defined and shown to generalize corresponding notions of graph connectivity. A theorem of Tutte on cyclically 3-connected graphs, is generalized to matroids.     

On Hamilton cycles in certain planar graphs

Let G be a 2-connected plane graph with outer cycle X_G such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of X_G. A sufficient condition for G to be Hamiltonian is presented. This theorem generalizes both Tutte's theorem that every 4-connected planar graph is Hamiltonian, as well as a recent theorem of Dillencourt about NST-triangulations. A linear algorithm to find a Hamilton cycle can be extracted from the proof. One corollary is that a 4-connected planar graph with the vertices of a triangle deleted is Hamiltonian. © 1996 John Wiley & Sons, Inc.     

A theorem on paths in planar graphs

We prove a theorem on paths with prescribed ends in a planar graph which extends Tutte's theorem on cycles in planar graphs [9] and implies the conjecture of Plummer [5] asserting that every 4-connected planar graph is Hamiltonian-connected.     

5连通图中的可去边

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.     

On Whitney's 2-isomorphism theorem for graphs

Let G and H be 2-connected 2-isomorphic graphs with n nodes. Whitney's 2-isomorphism theorem states that G may be transformed to a graph G* isomorphic to H by repeated application of a simple operation, which we will term “switching”. We present a proof of Whitney's theorem that is much shorter than the original one, using a graph decomposition by Tutte. The proof also establishes a surprisingly small upper bound, namely n-2, on the minimal number of switchings required to derive G* from G. The bound is sharp in the sense that for any integer N there exist graphs G and H with n ≥ N nodes for which the minimal number of switchings is n-2.     

More on betweenness-uniform graphs

We study graphs whose vertices possess the same value of betweenness centrality (which is defined as the sum of relative numbers of shortest paths passing through a given vertex). Extending previously known results of S. Gago, J. Hurajová, T. Madaras (2013), we show that, apart of cycles, such graphs cannot contain 2-valent vertices and, moreover, are 3-connected if their diameter is 2. In addition, we prove that the betweenness uniformity is satisfied in a wide graph family of semi-symmetric graphs, which enables us to construct a variety of nontrivial cubic betweenness-uniform graphs.     

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.     

Connectivity in bicircular matroids

Tutte has defined n-connection for matroids and proved a connected graph is n-connected if and only if its polygon matroid is n-connected. In this paper we introduce a new notion of connection in graphs, called n-biconnection, and prove an analogous theorem for graphs and their bicircular matroids. Results concerning 3-biconnected graphs are also presented.     

A construction of 3-connected graphs

We show that the 3-connected graphs can be generated from the complete graph on four vertices and the complete 3,3 bipartite graph by adding vertices and adding edges with endpoints on two edges meeting at a 3-valent vertex.     

Minimally 3-connected graphs

A constructive characterization of the class of minimally 3-connected graphs is presented. This yields a new characterization for the class of 3-connected graphs, which differs from the characterization provided by Tutte. Where Tutte's characterization requires the set of all wheels as a starting set, the new characterization requires only the graph K₄. The new characterization is based on the application of graph operations to appropriate vertex and edge sets in minimally 3-connected graphs.     

局部大边宽嵌入图的面圈和C-桥的结构

In this paper,we show that for a locally LEW-embedded 3-connected graph G in orientable surface,the following results hold:1) Each of such embeddings is minimum genus embedding;2) The facial cycles are precisely the induced nonseparating cycles which implies the uniqueness of such embeddings;3) Every overlap graph O(G,C) is a bipartite graph and G has only one C-bridge H such that CUH is nonplanar provided C is a contractible cycle shorter than every noncontractible cycle containing an edge of C.This ext...     

Intervals in matroid basis graphs

An interval in a graph is a subgraph induced by all the vertices on shortest paths between two given vertices. Intervals in matroid basis graphs satisfy many nice properties. Key results are: (1) any two vertices of a basis graph are together in some longest interval; (2) every basis graph with the minimum number of vertices for its diameter is an interval, indeed a hypercube. (1) turns out to be a simple case of a theorem in Edmonds' theory of matroid partition.     

Cycle Covers of Planar 2-Edge-Connected Graphs

A graph with n vertices is said to have a small cycle cover provided its edges can be covered with at most (2n ? 1)/3 cycles. Bondy [2] has conjectured that every 2-connected graph has a small cycle cover. In [3] Lai and Lai prove Bondy’s conjecture for plane triangulations. In [1] the author extends this result to all planar 3-connected graphs, by proving that they can be covered by at most (n + 1)/2 cycles. In this paper we show that Bondy’s conjecture holds for all planar 2-connected graphs. We also show that all planar 2-edge-connected graphs can be covered by at most (3n ? 3)/4 cycles and we show an infinite family of graphs for which this bound is attained.     

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 2-Connected Transmission Irregular Graphs

The transmission of a vertex v in a graph is the sum of the distances from v to all other vertices of the graph. In a transmission irregular graph, the transmissions of all vertices are pairwise distinct. It is known that almost all graphs are not transmission irregular. Some infinite family of transmission irregular trees was constructed by Alizadeh and Klav?ar [Appl.Math. Comput. 328, 113–118 (2018)] and the following problemwas formulated: Is there an infinite family of 2-connected graphs with the property? In this article, we construct an infinite family of 2-connected transmission irregular graphs.     

Cycles with Prescribed and Forbidden Sets of Elements in Cubic Graphs

 A necessary and sufficient condition for the existence of a cycle containing a set of three vertices and an edge excluding another edge is obtained for 3-connected cubic graphs by means of canonical contractions. A necessary and sufficient condition is also obtained for a cyclically 4-connected cubic graph to have a cycle that contains a set of four vertices and that avoids another vertex. Received: October 4, 1999 Final version received: June 14, 2000