Edward F. Schmeichel

On locating path- or tree-shaped facilities on networks

The study of “optimally” locating on a network a single facility of a given total length in the form of a path or a tree was initiated by several authors. We extend these results to the problem of locating p (≥1) such facilities. We will consider “center”, “median”, “max eccentricity”, and “max distance sum” location type problems for p = 1 or p > 1, for general networks and for tree networks, whether a facility contains partial arcs or not, and whether a facility is path-shaped or tree-shaped. These cases lead to 64 problems. We will determine the algorithmic complexity of virtually all these problems. We conclude with a result that may be viewed as a generalization of the p-Median theorem.

Recognizing tough graphs is NP-hard

We show that recognizing 1-tough graphs is NP-hard, thereby settling a long-standing open problem. We also prove it is NP-hard to recognize t-tough graphs for any fixed positive rational number t.

Toughness in graphs - A survey

In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. These categories are circumference, the disproof of the 2-tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness. We complete the survey with some tough open problems!

Long cycles in graphs with large degree sums

Abstract A number of results are established concerning long cycles in graphs with large degree sums. Let G be a graph on n vertices such that d(x)+d(y)+d(z)⩾s for all triples of independent vertices x, y, z. Let c be the length of a longest cycle in G and α the cardinality of a maximum independent set of vertices. If G is 1-tough and s⩾n, then every longest cycle in G is a dominating cycle and c⩾ min (n, n+ 1 3 s−α)⩾ min (n, 1 2 n+ 1 3 s)⩾ 5 6 n . If G is 2-connected and s⩾n+2, then also c⩾ min (n, n+ 1 3 s-α) , generalizing a result of Bondy and one of Nash-Williams. Finally, if G is 2-tough and s⩾n, then G is hamiltonian.

A cycle structure theorem for Hamiltonian graphs

An n-vertex graph is called pancyclic if it contains a cycle of length 1 for every 1 such that 3 n. Then G is either (i) pancyclic, (ii) bipartite, or (iii) missing only an (n I)-cycle. Moreover, (iii) can occur only if G has a very explicit structure “near” x and y. This result can be used to show that three well-known hamiltonian degree conditions (due to Chvatal, Fan, and Bondy) actually imply that a graph is essentially pancyclic.

The Voronoi partition of a network and its implications in location theory

Given a network N(V, E) and a set of points Xp = {x1, …, xp} on N, we first present an algorithm for computing the Voronoi partition of N(V, E) into territories T(x1), …, T(xp). After describing two ways to measure the “size” of a territory, we introduce and discuss the more challenging problem of selecting Xp so that the maximum size among the resulting territories is as small as possible. For one especially natural way to measure the size of a territory, we show that this latter problem is NP-complete when p is part of the input, but that the problem can be solved in polynomial time for any fixed p. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

An adaptive algorithm for system level diagnosis

Abstract Consider a system of n units, at most t of which are faulty. An adaptive diagnosis algorithm is presented which uses a sequence of tests to identify a fault-free unit. The algorithm requires at most 2 t − ν ( t ) tests, where ν ( t ) is the number of 1s in the binary representation of t . Moreover, many of the tests can be performed simultaneously. The previously best algorithms for the same purpose required 2 t − 1 tests, none of which could be performed simultaneously.

Orienting graphs to optimize reachability

It is well known that every 2-edge-connected graph can be oriented so that the resulting digraph is strongly connected. Here we study the problem of orienting a connected graph with cut edges in order to maximize the number of ordered vertex pairs (x, y) such that there is a directed path from x to y. After transforming this problem, we prove a key theorem about the transformed problem that allows us to obtain a quadratic algorithm for the original orientation problem. We also consider how to orient graphs to minimize the number of ordered vertex pairs joined by a directed path. After showing this problem is equivalent to the comparability graph completion problem, we show both problems are NP-hard, and even NP-hard to approximate to within a factor of 1 + e, for some e > 0.

Star arboricity of graphs

Abstract We develop a connection between vertex coloring in graphs and star arboricity which allows us to prove that every planar graph has star arboricity at most 5. This settles an open problem raised independently by Algor and Alon and by Ringel. We also show that deciding if a graph has star arboricity 2 is NP-complete, even for 2-degenerate graphs.

Bipartite graphs with cycles of all even lengths

Let G = (X, Y, E) be a bipartite graph with X = Y = n. Chvatal gave a condition on the vertex degrees of X and Y which implies that G contains a Hamiltonian cycle. It is proved here that this condition also implies that G contains cycles of every even length when n > 3.