Jørgen Bang-Jensen

When the greedy algorithm fails

We provide a characterization of the cases when the greedy algorithm may produce the unique worst possible solution for the problem of finding a minimum weight base in an independence system when the weights are taken from a finite range. We apply this theorem to TSP and the minimum bisection problem. The practical message of this paper is that the greedy algorithm should be used with great care, since for many optimization problems its usage seems impractical even for generating a starting solution (that will be improved by a local search or another heuristic).

Quasi-transitive digraphs

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interseting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, we show that every strongly connected quasi-transitive digraphs D on at least four vertices has two vertices v1 and v2 such that D – vi is strongly connected for i = 1, 2. A result of tournaments on the existence of a pair of arc-disjoint in- and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore we characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 ≦ k ≦ |V(D)| − 1 through every vertex and yet they are not hamiltonian. Finally we characterize pancyclic and vertex pancyclic quasi-transitive digraphs.

The effect of two cycles on the complexity of colouring by directed graphs

Abstract Let H be a fixed directed graph. An H-colouring of a directed graph D is a mapping f : V ( D )→ V ( H ) such that f ( x ) f ( y ) is an edge of H whenever xy is an edge of D . We study the following H-colouring problem : Instance : A directed graph D . Question : Does there exist an H-colouring of D ? In an earlier paper [2] it is shown that among semicomplete digraphs H , it is the existence of two directed cycles in H which makes the H -colouring problem (NP-) hard. In this paper we provide further classes of digraphs in which two directed cycles in H make the H -colouring problem NP-hard. These include both classes of dense and of sparse digraphs. There still appears to be no natural conjecture as to what digraphs H give NP-hard H -colouring problems; however, in view of our results, we are led to make such a conjecture for digraphs without sources and sinks.

A polynomial algorithm for the 2-path problem for semicomplete digraphs

This paper presents polynomially bounded algorithms for finding a cycle through any two prescribed arcs in a semicomplete digraph and for finding a cycle through any two prescribed vertices in a complete k-partite oriented graph. It is also shown that the problem of finding a maximum transitive subtournament of a tournament and the problem of finding a cycle through a prescribed arc set in a tournament are both NP-complete.

Preserving and Increasing Local Edge-Connectivity in Mixed Graphs

Generalizing and unifying earlier results of W. Mader, and A. Frank and B. Jackson, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain min-max formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satisfies local edge-connectivity prescriptions. An extension of Edmondss theorem on disjoint arborescences is also deduced along with a new sufficient condition for the solvability of the edge-disjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems.

In-tournament digraphs

Abstract In this paper we introduce a generalization of digraphs that are locally tournaments (and hence of tournaments). This is the class of in-tournament digraphs-the set of predecessors of every vertex induces a tournament. We show that many properties of local tournament digraphs can be extended even to in-tournament digraphs. For instance, any strongly connected in-tournament digraph has a directed hamiltonian cycle. We prove that the underlying graph of any in-tournament digraph is l-homotopic. We investigate the problem of which graphs are orientable as in-tournament digraphs and prove that any graph representable as an intersection subgraph of a unicyclic graph can be so oriented. It is shown that there is a polynomial algorithm for recognizing those graphs that can be oriented as in-tournament digraphs.

Alternating cycles and paths in edge-coloured multigraphs: a survey

Abstract A path or cycle in an edge-coloured multigraph is called alternating if its successive edges differ in colour. We survey results of both theoretical and algorithmic character concerning alternating cycles and paths in edge-coloured multigraphs. We also show useful connections between the theory of paths and cycles in bipartite digraphs and the theory of alternating paths and cycles in edge-coloured graphs.

Edge-Connectivity Augmentation with Partition Constraints

In the well-solved edge-connectivity augmentation problem we must find a minimum cardinality set F of edges to add to a given undirected graph to make it k-edge-connected. This paper solves the generalization where every edge of F must go between two different sets of a given partition of the vertex set. A special case of this partition-constrained problem, previously unsolved, is increasing the edge-connectivity of a bipartite graph to k while preserving bipartiteness. Based on this special case we present an application of our results in statics. Our solution to the general partition-constrained problem gives a min-max formula for |F| which includes as a special case the original min-max formula of Cai and Sun [Networks, 19 (1989), pp. 151--172] for the problem without partition constraints. When k is even the min-max formula for the partition-constrained problem is a natural generalization of the unconstrained version. However, this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge. We give a strongly polynomial algorithm that solves our problem in time O(n(m + nlog n)log n). Here n and m denote the number of vertices and distinct edges of the given graph, respectively. This bound is identical to the best-known time bound for the problem without partition constraints. Our algorithm is based on the splitting off technique of Lovasz, like several known efficient algorithms for the unconstrained problem. However, unlike previous splitting algorithms, when k is odd our algorithm must handle obstacles that prevent all edges from being split off. Our algorithm is of interest even when specialized to the unconstrained problem, because it produces an asymptotically optimum number of distinct splits.

Local Tournaments and Proper Circular Arc Gaphs

A local tournament is a digraph in which the out-set as well as the in-set of every vertex is a tournament. These digraphs have recently been found to share many desirable properties of tournaments. We illustrate this by giving O(m+n logn) algorithms to find a hamiltonian path and cycle in a local tournament. We mention several characterizations and recognition algorithms of graphs orientable as local tournaments. It turns out that they are precisely the graphs previously studied as proper circular arc graphs. Thus we obtain new recognition algorithms for proper circular arc graphs. We also give a more detailed structural characterization of chordal graphs that are orientable as local tournaments, i.e., that are proper circular arc graphs.

A sufficient condition for a semicomplete multipartite digraph to be Hamiltonian

Abstract A multipartite tournament is an orientation of a complete k-partite graph for some k ⩾ 2. A factor of a digraph D is a collection of vertex disjoint cycles covering all the vertices of D. We show that there is no degree of strong connectivity which together with the existence of a factor will guarantee that a multipartite tournament is Hamiltonian. Our main result is a sufficient condition for a multipartite tournament to be Hamiltonian. We show that this condition is general enough to provide easy proofs of many existing results on paths and cycles in multipartite tournaments. Using this condition, we obtain a best possible lower bound on the length of a longest cycle in any strongly connected multipartite tournament.