Extremal Graph Theoretic Problems with Applications

Abstract

Graphs have now become recognized models for a wide variety of situations. Whenever we have a collection of objects with a binary relation defined on them graphs serve as excellent tools to study the combinatorial properties of the collection with respect to the binary relation. D. Konige was the first person to recognize the usefulness of graph theoretic models. He conceived of a unified study of graphs under an abstract set up. His book was a pioneering work in this field. The graph theoretic problems embodied in this thesis have been motivated by situations arising in communication networks. In terms of graphs these problems ask for the extremal structures with preassigned diameters and their variations under suppression of vertices and edges. The motivation for the problems and their applications is deferred to the penultimate chapter. This has been found reasonable, in any case not disorderly, as the appropriate combinatorial problems in terms of graphs seem to be of great interest on their own. Perhaps we have the cue here to study the extremal structures of graphs satisfying a given property and retaining the same property after some portions of the graphs have been suppressed. All the graph theoretic problems considered here centre round the distance metric defined for graphs. An attempt is made in the last chapter to define a distance between any two columns of a (0, 1)-matrix. In light of this it seems to be possible to carry the analogy from graphs to matrices. The contributions of this thesis have been divided into six chapters and an appendix. At the beginning of each chapter is provided. Berge [1] has been followed throughout for notation and terminology.