Diametral Path Graphs and Incremental Distance Sequences.

Abstract

Summary of Research Our research efforts funded by ONR have produced excellent results in several different areas of graph theory and its applications. Hamiltonian Graphs: In Hamiltonian graphs our approach was twofold. On one hand we investigated sufficient degree and edge conditions for balanced bipartite graphs to be hamiltonian. On the other hand, we developed algorithms for finding Hamiltonian paths and cycles in permutation and cocomparability graphs. It may be noted that Hamiltonian cycle problem was well known open problem since 1985. We developed 0(n 2 ) algorithm for permutation graphs and 0{n z ) algorithm for cocomparability graphs. In addition, toughness properties of permutation graphs and cocomparability graphs were also investigated. These results contribute significantly to the understanding of Hamiltonian properties. Line Graphs and their generalizations: Line graphs provide a way of studying the graph by concentrating attention on edges without regard to vertices. We generalized the notion of fine graphs to super line graphs and obtained several results about their properties. Our approach studies fine graphs combinatorially, by looking at sets of edges of a given cardinality. Several interresting new parameters related to the notion of super fine graphs have been introduced and studied. This study contributes significantly to the generalizations of the line graph transformation.