Recent results on well-balanced orientations

In this paper we consider problems related to Nash-Williams´ Strong Orientation Theorem and Odd-Vertex Pairing Theorem. These theorems date to 1960 and up to now not much is known about their relationship to other subjects in graph theory. We investigated many approaches to find a more transparent proof for these theorems and possibly generalizations of them. In many cases we found negative answers: counter-examples and NP-completeness results. For example we show that the weighted and the degree-constrained versions of the well-balanced orientation problem are NP-hard. We also show that it is NP-hard to find a minimum cost feasible odd-vertex pairing or to decide whether two graphs with some common edges have simultaneous well-balanced orientations or not. Nash-Williams´ original approach was to define best-balanced orientations with feasible odd-vertex pairings: we show here that not every best-balanced orientation can be obtained this way. However we prove that in the global case this is true: every smooth k-arc-connected orientation can be obtained through a k-feasible odd-vertex pairing. The aim of this paper is to help to find a transparent proof for the Strong Orientation Theorem. In order to achieve this we propose some other approaches and raise some open questions, too. (c) 2008 Elsevier B.V. All rights reserved

Checking the admissibility of odd-vertex pairings is hard

Nash-Williams proved that every graph has a well-balanced orientation. A key ingredient in his proof is admissible odd-vertex pairings. We show that for two slightly different definitions of admissible odd-vertex pairings, deciding whether a given odd-vertex pairing is admissible is co-NP-complete. This resolves a question of Frank. We also show that deciding whether a given graph has an orientation that satisfies arbitrary local arc-connectivity requirements is NP-complete

Orientations des graphes (structures et algorithmes)

Orienter un graphe c'est remplacer chaque arête par un arc de mêmes extrémités. On s'intéresse à la connexité du graphe orienté ainsi obtenu. L'orientation avec des contraintes d'arc-connexité est maintenant comprise en profondeur mais très peu de résultats sont connus en terme de sommet-connexité. La conjecture de Thomassen avance que les graphes suffisament sommet-connexes ont une orientation k-sommet-connexe. De plus, la conjecture de Frank propose une caractérisation des graphes qui admettent une telle orientation. Les résultats de cette thèse s'articulent autour des notions d'orientation, de packing, de connexité et de matroïde. D'abord, nous infirmons une conjecture de Recski sur la décomposition d'un graphe en arbres ayant des orientations avec degrés entrants prescrits. Nous prouvons également un nouveau résultat sur le packing d'arborescences enracinées avec contraintes de matroïdes. Ceci généralise un résultat fondamental d'Edmonds. Enfin, nous démontrons un nouveau théorème de packing sur les bases des matroïdes de dénombrement qui nous permet d'améliorez le seul résultat connu sur la conjecture de Thomassen. D'autre part, nous donnons une construction et un théorème d'augmentation pour une famille de graphes liée à la conjecture de Frank. En conclusion, nous réfutons la conjecture de Frank et prouvons que, pour tout entier k >= 3, décider si un graphe a une orientation k-sommet-connexe est un problème NP-complet.Orienting an undirected graph means replacing each edge by an arc with the same ends. We investigate the connectivity of the resulting directed graph. Orientations with arc-connectivity constraints are now deeply understood but very few results are known in terms of vertex-connectivity. Thomassen conjectured that sufficiently highly vertex-connected graphs have a k-vertex- connected orientation while Frank conjectured a characterization of the graphs admitting such an orientation. The results of this thesis are structures around the concepts of orientation, packing, connectivity and matroid. First, we disprove a conjecture of Recski on decomposing a graph into trees having orientations with specified indegrees. We also prove a new result on packing rooted arborescences with matroid constraints. This generalizes a fundamental result of Edmonds. Moreover, we show a new packing theorem for the bases of count matroids that induces an improvement of the only known result on Thomassen's conjecture. Secondly, we give a construction and an augmentation theorem for a family of graphs related to Frank's conjecture. To conclude, we disprove the conjecture of Frank and prove that, for every integer k >= 3, the problem of deciding whether a graph admits a k-vertex-orientation is NP-complete.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Recent results on well-balanced orientations

International audienc

Recent Results on Well-Balanced Orientations

In this paper we consider problems related to Nash-Williams' Strong Orientation Theorem and Odd-Vertex Pairing Theorem. These theorems date to 1960 and up to now not much is known about their relationship to other subjects in graph theory. We investigated many approaches to find a more transparent proof for these theorems and possibly generalizations of them. In many cases we found negative answers: counter-examples and NP-completeness results. For example we show that the weighted and the degree-constrained versions of the well-balanced orientation problem are NP-hard. We also show that it is NP-hard to find a minimum cost feasible odd-vertex pairing or to decide whether two graphs with some common edges have simultaneous well-balanced orientations or not. Nash-Williams' original approach was to define best-balanced orientations with feasible odd-vertex pairings: we show here that not every best-balanced orientation can be obtained this way. However we prove that in the global case this is true: every smooth k-arc-connected orientation can be obtained through a k-feasible odd-vertex pairing. The aim of this paper is to help to find a transparent proof for the Strong Orientation Theorem. In order to achieve this we propose some other approaches and raise some open questions, too. (c) 2008 Elsevier B.V. All rights reserved