Trouver de bonnes 2-partitions des digraphes I. Propriétés héréditaires.

We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote following two sets of natural properties of digraphs: H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E ={strongly connected, connected, minimum out- degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper, we determine the complexity of deciding, for any fixed pair of positive integers k1,k2, whether a given digraph has a vertex partition into two digraphs D1,D2 such that |V(Di)| ≥ ki and Di has property Pi for i = 1, 2 when P1 ∈ H and P2 ∈ H ∪ E. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the problems when both P1 and P2 are in E is determined in the companion paper [2].when both $\mathbb{P}_1$ and $\mathbb{P}_2$ are in ${\cal E}$ is determined in the companion paper (INRIA Research report RR-8868).Nous étudions la complexité de décider si un digraphe donné D admet une partition en deux sous-digraphes ayant des propriétés structurelles fixées. Dénotons par H et E les deux ensembles de propriétés de digraphes naturelles : H ={acyclique, complet, sans arcs, orienté, semicomplet, symétrique, tournoi} et E ={fortement connexe, connexe, degré sortant minimum au moins 1, degré entrant minimum au moins 1, semi-degré entrant minimum au moins 1, degré minimum au moins 1, avoir une arborescence sortante couvrante, avoir une arborescence entrante couvrante}. Dans ce rapport, nous déterminons la complexité de décider, pour toute paire d’entiers k1,k2, si un digraphie donné admet une partition en deux digraphes D1,D2 tels que |V(Di)|≥ki et Di a la propriété Pi pour i=1,2lorsque P1 ∈H et P2 ∈H∪E. Nous classifions également la complexité des mêmes problèmes restreints aux digraphies fortement connexes. La complexité des problèmes lorsque P1 et P2 sont toutes deux dans E est déterminée dans le rapport suivant (Rapport de Recherche INRIA RR-8868)

Finding good 2-partitions of digraphs I. Hereditary properties

International audienceWe study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote following two sets of natural properties of digraphs: H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E ={strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper, we determine the complexity of of deciding, for any fixed pair of positive integers k1, k2, whether a given digraph has a vertex partition into two digraphs D1, D2 such that |V (Di)| ≥ ki and Di has property Pi for i = 1, 2 when P1 ∈ H and P2 ∈ H ∪ E. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the problems when both P1 and P2 are in E is determined in the companion paper [2]

The Complexity of Surjective Homomorphism Problems -- a Survey

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of open complexity

Matter-antimatter asymmetry restrains the dimensionality of neural representations: quantum decryption of large-scale neural coding

Projections from the study of the human universe onto the study of the self-organizing brain are herein leveraged to address certain concerns raised in latest neuroscience research, namely (i) the extent to which neural codes are multidimensional; (ii) the functional role of neural dark matter; (iii) the challenge to traditional model frameworks posed by the needs for accurate interpretation of large-scale neural recordings linking brain and behavior. On the grounds of (hyper-)self-duality under (hyper-)mirror supersymmetry, inter-relativistic quantum principles are introduced, whose consolidation, as spin-geometrical pillars of a network- and game-theoretical construction, is conducive to (i) the high-precision reproduction and reinterpretation of core experimental observations on neural coding in the self-organizing brain, with the instantaneous geometric dimensionality of neural representations of a spontaneous behavioral state being proven to be at most 16, unidirectionally; (ii) a possible role for spinor (co-)representations, as the latent building blocks of self-organizing cortical circuits subserving (co-)behavioral states; (iii) an early crystallization of pertinent multidimensional synaptic (co-)architectures, whereby Lorentz (co-)partitions are in principle verifiable; and, ultimately, (iv) potentially inverse insights into matter-antimatter asymmetry. New avenues for the decryption of large-scale neural coding in health and disease are being discussed.Comment: 33 pages;3 figures;1 table;minor edit

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

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