Showing papers on "Longest path problem published in 1971"

Steiner's problem in graphs and its implications

Abstract: A graph theoretic version of Steiner's problem in plane geometry is described. An approach for solving this problem, related to Melzak's solution to Steiner's problem, is presented. The problems of finding “shortest route” and “minimal spanning tree” in graphs become special cases of the Steiner's problem in graphs. It is shown that a solution to this problem also provides us with a solution to the problems of finding a minimum externally stable set and a maximum internally stable set in a graph.

Paths and Cycles in Digraphs

Abstract: This chapter illustrates the paths and cycles in digraphs. From Warshall's theorem, if X is the adjacency matrix of a digraph, then the X* generated by the set j = j + 1, when j ≧ n is the path matrix. The chapter discusses the shortest path problems. A road map is a labeled graph. If two towns in a densely populated area are three or four hundred miles apart, very many routes lead from one to the other. The one with the least total mileage is the shortest path. Finding the shortest path is a very real practical problem. The path that is shortest in the sense of least traveling time is not always the path of shortest distance; it may take twice as long to cover 10 miles on a country lane as to go 20 miles on a freeway. The shortest path given by the general algorithm must, then, be the path of fewest arcs. The chapter presents specifications of all the simple paths and cycles in a digraph. The algorithm is based on a matrix operation called symbolic multiplication.