The competition graph of a symmetric digraph D with a loop at each vertex is the square of the underlying graph of D with loops removed. In the interest of eeciently assigning radio frequencies in a communication network, we would like to determine which graphs have interval squares. Necessary and suucient conditions on the graph G are given for G 2 to be interval for several large classes of g...

Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through t...

Given a tree and a set of paths in the tree, the problem of finding a minimum number of paths from the given path set to cover all the vertices in the tree is investigated in the paper. To distinguish from the classical path cover problem, such an optimization problem is referred to as vertex covering by paths. The problem and its edge variant, edge covering by paths, find applications in machi...

The Push-Relabel, or Preflow-Push, Maximum-Flow algorithm was originally developed by Andrew V. Goldberg and Robert E. Tarjan in the mid-to-late 1980s, [2, 3]. This Push-Relabel algorithm is unlike the previous Ford-Fulkerson and Edmonds-Karp algorithms which use the concept of an augmenting path to determine the maximum flow. This algorithm instead uses a concept of a preflow, originally devel...

In the last four decades, combinatorial optimization has been strongly influenced by linear programming. With the mathematical and algorithmic understanding of linear programs came a whole host of ideas and tools that were then applied to combinatorial optimization. Many of these ideas and tools are still in use today, and form the bedrock of our understanding of combinatorial optimization. One...

We attempt to generalize a theorem of Nash-Williams stating that graph has k-arc-connected orientation if and only it is 2k-edge-connected. In strongly connected digraph we call an arc deletable its deletion leaves digraph. Given 3-edge-connected G, define Frank number f(G) be the minimum k such there exist orientations G with property every edge becomes in at least one these orientations. are ...

This paper presents some modifications of Ford-Fulkerson’s labeling method for solving the maximal network flow problem with application in solving the transportation and assignment problems. The modifications involve the tree representation of the nodes labeled and the edges used them. It is shown that after each flow adjustment some of the labels can be retained for the next labeling process....

1.1. Intervals. We shall denote by I a finite set of closed intervals of the real line. Investigations of combinatorial properties of intervals started by Gy. Hajós (1957, [40]) T. Gallai ([26]) S. Benzer (1959, [8]) P. C. Gilmore and A. J. Hoffman (1964, [7]) D. R. Fulkerson and O. A. Gross (1965, [22]) C. G. Lekkerkerker and J. Ch. Boland (1962, [57]). It is traditional to display a set of in...