Topics in Structural Graph Theory by Lowell W. Beineke, Robin J. Wilson, Ortrud R. Oellermann

By Lowell W. Beineke, Robin J. Wilson, Ortrud R. Oellermann

The speedily increasing sector of structural graph concept makes use of principles of connectivity to discover quite a few points of graph concept and vice versa. It has hyperlinks with different parts of arithmetic, corresponding to layout conception and is more and more utilized in such parts as desktop networks the place connectivity algorithms are a tremendous function. even though different books disguise elements of this fabric, none has a equally vast scope. Ortrud R. Oellermann (Winnipeg), across the world known for her massive contributions to structural graph thought, acted as educational advisor for this quantity, assisting form its insurance of key themes. the result's a set of 13 expository chapters, every one written by way of stated specialists. those contributions were conscientiously edited to reinforce clarity and to standardise the bankruptcy constitution, terminology and notation all through. An introductory bankruptcy information the history fabric in graph concept and community flows and every bankruptcy concludes with an in depth checklist of references.

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Aharoni strengthened their result when he showed the conjecture to be true for all graphs that contain no infinite path [1] and for all countable graphs [2]. 3. Edge-connectivity The vertex versions of Menger’s theorem discussed in Section 2 have edge analogues that we briefly describe here. Let v and w be two vertices in a graph G. A set S of edges is a v–w edge-separating set if v and w lie in different components of G − S: that is, if every v–w path contains an edge of S. The minimum cardinality of a v–w edge-separating set is the v–w edge-connectivity and is denoted by λ(v, w).

6. F. T. Boesch and S. Chen, A generalization of line connectivity and optimally invulnerable graphs, SIAM J. Appl. Math. 34 (1978), 657–665. 7. S. M. Boyles and G. Exoo, A counterexample on a conjecture on paths of bounded lengths, J. Graph Theory 6 (1982), 205–209. 8. G. Chartrand, A graph theoretic approach to a communications problem, SIAM J. Appl. Math. 14 (1996), 778–781. 9. G. Chartrand and F. Harary, Graphs with prescribed connectivities, Theory of Graphs, Proceedings of the Colloquium Held at Tihany, Hungary, Akadémiai Kaidó, 1968.

A graph G is uniformly k-connected if κ(G) = κ(G) = k. A graph G with connectivity k is critically k-connected if κ(G − v) < k for each vertex v, and is minimally k-connected if κ(G−e) < k for each edge e. Some necessary conditions for a graph to be uniformly k-connected were given by Beineke, Oellermann and Pippert [5]. 2 Every uniformly k-connected graph is minimally k-connected if k ≥ 1, and is critically k-connected if k ≥ 2. These conditions are not sufficient, as there are graphs that are both minimally and critically k-connected, but not uniformly k-connected.

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