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Báo cáo toán học: "The Structure of Locally Finite Two-Connected Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The Structure of Locally Finite Two-Connected Graphs. | The Structure of Locally Finite Two-Connected Graphs Carl Droms1 Brigitte Servatius and Herman Servatius Submitted May 15 1995 Accepted September 4 1995. Abstract We expand on Tutte s theory of 3-blocks for 2-connected graphs generalizing it to apply to infinite locally finite graphs and giving necessary and sufficient conditions for a labeled tree to be the 3-block tree of a 2-connected graph. Mathematics Subject Classification 05C40 05C38 and 05C05. Key Words and Phrases Tutte connectivity 3-block 3-block tree 1 Introduction Connectivity properties of graphs are among the basic aspects of graph theory. Every graph is the disjoint union of its connected components and every connected graph is the edge disjoint union of its maximal 2-connected subgraphs encoded in the block-cutpoint tree. A canonical decomposition for finite 2-connected graphs was given by Tutte 11 in the form of the 3-block tree and generalized to matroids by Cunningham and Edmonds 1 . Such decompositions are important tools in inductive arguments and constructions. Hopcroft and Tarjan 4 gave an important algorithm for computing the 3-block tree of a graph in O V E time which is comparable to the complexity of computing other non-canonical decompositions say the ear decomposition and is also applicable to matroids. Effective decompositon schemes for graphs of connectivity 3 and higher have been given but none are canonical and in Section 6 we argue that none will be forthcoming. The uniqueness of Tutte s construction may be exploited to study the symmetry properties of graphs with low connectivity 8 and 2 particularly in the case of planar graphs 3 . In this paper we will examine the interpretation of Tutte s decomposition and extend the theory to infinite graphs. 2 n-Connectivity We are concerned with the structure of locally finite graphs of connectivity less than 4 allowing graphs to have loops and multiple edges. If a graph G has at least 3 vertices and is not a triangle then G is defined to .