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]

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)

The reversing number of a diagraph

AbstractA minimum reversing set of a diagraph is a smallest sized set of arcs which when reversed makes the diagraph acyclic. We investigate a related issue: Given an acyclic diagraph D, what is the size of a smallest tournament T which has the arc set of D as a minimun reversing set? We show that such a T always exists and define the reversing number of an acyclic diagraph to be the number of vertices in T minus the number of vertices in D. We also derive bounds and exact values of the reversing number for certain classes of acyclic diagraphs

Decomposing tournaments into paths

We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by Kühn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament T . There is a natural lower bound for this number in terms of the degree sequence of T and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them

On width measures and topological problems on semi-complete digraphs

Under embargo until: 2021-02-01The topological theory for semi-complete digraphs, pioneered by Chudnovsky, Fradkin, Kim, Scott, and Seymour [10], [11], [12], [28], [43], [39], concentrates on the interplay between the most important width measures — cutwidth and pathwidth — and containment relations like topological/minor containment or immersion. We propose a new approach to this theory that is based on outdegree orderings and new families of obstacles for cutwidth and pathwidth. Using the new approach we are able to reprove the most important known results in a unified and simplified manner, as well as provide multiple improvements. Notably, we obtain a number of efficient approximation and fixed-parameter tractable algorithms for computing width measures of semi-complete digraphs, as well as fast fixed-parameter tractable algorithms for testing containment relations in the semi-complete setting. As a direct corollary of our work, we also derive explicit and essentially tight bounds on duality relations between width parameters and containment orderings in semi-complete digraphs.acceptedVersio

Sweeping Graphs and Digraphs

Searching a network for an intruder is an interesting and difficult problem. Sweeping is one such search model, in which we "sweep" for intruders along edges. The minimum number of sweepers needed to clear a graph G is known as the sweep number sw(G). The sweep number can be restricted by insisting the sweep be monotonic (once an edge is cleared, it must stay cleared) and connected (new clear edges must be incident with already cleared edges). We will examine several lower bounds for sweep number, among them minimum degree, clique number, chromatic number, and girth. We will make use of several of these bounds to calculate sweep numbers for several infinite families of graphs. In particular, these families will answer some open problems regarding the relationships between the monotonic sweep number, connected sweep number, and monotonic connected sweep number. While sweeping originated in simple graphs, the idea may be easily extended to directed graphs, which allow for four different sweep models. We will examine some interesting non-intuitive sweep numbers and look at relations between these models. We also look at bounds on these sweep numbers on digraphs and tournaments