On packing time-respecting arborescences

We present a slight generalization of the result of Kamiyama and Kawase (2015) on packing time-respecting arborescences in acyclic pre-flow temporal networks. Our main contribution is to provide the first results on packing time-respecting arborescences in non-acyclic temporal networks. As negative results, we prove the NP-completeness of the decision problem of the existence of 2 arc-disjoint spanning time-respecting arborescences and of a related problem proposed in this paper.     

Arc-disjoint arborescences of digraphs

The purpose of this paper is to give a necessary and sufficient condition for a digraph G to contain k arc-disjoint arborescences so that the number rooted at each vertex x of G lies in some prescribed interval which depends on x.     

A note on disjoint arborescences

Recently Kamiyama, Katoh, and Takizawa have shown a theorem on packing arc-disjoint arborescences that is a proper extension of Edmonds’ theorem on disjoint spanning branchings. We show a further extension of their theorem, which makes clear an essential rôle of a reachability condition played in the theorem. The right concept required for the further extension is “convexity” instead of “reachability”.     

Exact arborescences,matchings and cycles

Suppose we are given a graph in which edge has an integral weight. An ‘exact’ problem is to determine whether a desired structure exists for which the sum of the edge weights is exactly k for some prescribed k.We consider the special case of the problem in which all costs are zero or one for arborescences and show that a ‘continuity’ property is prossessed similar to that possessed by matroids. This enables us to determine in polynomial time the complete set of values of k for which a solution exists. We also give a minmax theorem for the maximum possible value of k, in terms of a packing of certain directed cuts in the graph.We also show how enumerative techniques can be used to solve the general exact problem for arborescences (implying spanning trees), perfect matchings in planar graphs and sets of disjoint cycles in a class of planar directed graphs which includes those of degree three. For these problems, we thereby obtain polynomial algorithms provided that the weights are bounded by a constant or encoded in unary.     

Minimax regret spanning arborescences under uncertain costs

The paper considers a classical optimization problem on a network whose arc costs are partially known. It is assumed that an interval estimate is given for each arc cost and no further information about the statistical distribution of the truth value of the arc cost is known. In this context, given a spanning arborescence in the network, its cost can take on different values according to the choice of each individual arc cost, that is, according to the different cost scenarios.     

Sur les arborescences dans un graphe oriente

The main result of this paper is the following theorem: Let G = (X,E) be a digraph without loops or multiple edges, |X| ?3, and h be an integer ?1, if G contains a spanning arborescence and if d⁺_G(x)+d^?_G(x)+d^?_G(y)+d^?_G(y)? 2|X |?2h?1 for all x, y?X, x ≠ y, non adjacent in G, then G contains a spanning arborescence with ?h terminal vertices. A strengthening of Gallai-Milgram's theorem is also proved.     

Spanning arborescences,ingraphs, and outgraphs

An ingraph N is a subgraph of a digraph G whose edge set consists of all the edges of G that are directed into a subset X of the vertices. Set X is the generating set of N. It is proved that G contains a unique even ingraph and this ingraph is generated by the set A of vertices that root an odd number of spanning out arborescences provided A is nonempty. If A is empty, then there exist at least two even ingraphs.     

The maximum number of arc-disjoint arborescences in a tournament

In this article, we give the maximum number of arc-disjoint arborescences in a tournament or an oriented complete r-partite graph by means of the indegrees of its vertices.     

Line-digraphs,arborescences, and theorems of tutte and knuth

The line-digraph of a digraph D with vertices V₁, …, V_n is the digraph $\text{D}{}^1$ obtained from D by associating with each edge of D a vertex of $\text{D}{}^1$, and then directing an edge from vertex (V_i, V_j) of $\text{D}{}^1$ to vertex (V_k, V_m) if and only if j = k. This paper extends a characterization given by Harary and Norman for linedigraphs. It is also possible to repeatedly contract vertices of the line-digraph (with a new contraction procedure) so as to obtain the digraph derived from D by deleting all vertices with no incoming edges. Several new identities for arborescences are presented, leading to a combinatorial proof of Knuth's formula for the number of arborescences of a line-digraph. A new proof is given for the fact that in a digraph with every vertex having indegree equal to outdegree, the number of arborescences with root V_j is independent of j. Finally a new proof is presented for Tutte's Matrix Tree Theorem which shows the theorem to be a special case of the principle of inclusion-exclusion.     

Enumerative problems for arborescences and monotone paths on polytope graphs

Every generic linear functional $f$ on a convex polytope $P$ induces an orientation on the graph of $P$. From the resulting directed graph one can define a notion of $f$-arborescence and $f$-monotone path on $P$, as well as a natural graph structure on the vertex set of $f$-monotone paths. These concepts are important in geometric combinatorics and optimization. This paper bounds the number of $f$-arborescences, the number of $f$-monotone paths, and the diameter of the graph of $f$-monotone paths for polytopes $P$ in terms of their dimension and number of vertices or facets.     

Packing of arborescences with matroid constraints via matroid intersection

Mathematical Programming - Edmonds’ arborescence packing theorem characterizes directed graphs that have arc-disjoint spanning arborescences in terms of connectivity. Later he also observed a...     

Packing algorithms for arborescences (and spanning trees) in capacitated graphs

In a digraph with real-valued edge capacities, we pack the greatest number of arborescences in time O(n ³m log(n ²/m)); the packing uses at mostm distinct arborescences. Heren andm denote the number of vertices and edges in the given graph, respectively. Similar results hold for integral packing: we pack the greatest number of arborescences in time O(min{n, logN}n ²m log(n ²/)) using at mostm + n – 2 distinct arborescences; hereN denotes the largest (integral) capacity of an edge. These resuts improve the best previous bounds for capacitated digraphs. The algorithm extends to several related problems, including packing spanning trees in an undirected capacitated graph, and covering such graphs by forests. The algorithm provides a new proof of Edmonds' theorem for arborescence packing, for both integral and real capacities, based on a laminar family of sets derived from the packing. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Blocking sets

We introduce techniques to obtain some new results on blocking sets and to extend some known ones.     

On the complexity of some arborescences finding problems on a multihop radio network

An arborescence of a multihop radio network is a directed spanning tree (with rootx) such that the edges are directed away from the root. Based upon an arborescence,x canbroadcast a message to other nodes according to the directed edges of the spanning tree. The minimum transmission power arborescence problem is to find an arborescence such that the message can be broadcasted to other nodes by using a minimal amount of transmission power. The minimum delay arborescence problem is to find an arborescence such that a message can be broadcasted to other nodes by using a minimal number of broadcast transmission. In this paper we show that both these problems areNP-complete. The reductions are from the maximum leaf spanning tree problem.Areverse arborescence is similar to an arborescence except that the edges are directed toward the root. Based upon a reverse arborescence, the root node cancollect information from other nodes. In this paper we also show that the reverse minimum transmission power arborescence problem can be solved with the same computational complexity as that of finding a minimum cost spanning tree, and the reverse minimum delay arborescence problem can be solved with the same computational complexity as that of finding a spanning tree.