We study the shortest-path broadcast problem in graphs and digraphs, where amessage has to be transmitted from a source node s to all the nodes along shortest paths, in the classical telephonemodel. For both graphs and digraphs, we show that the problem is equivalent to the broadcast problem in layered directed graphs. We then prove that this latter problem is NP-hard, and therefore that the sh...

We consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of “compatible Euler tours” or “orthogonal A-trails” by Andsersen at al [1] and by Bouchet [4]. This prior work has mostly been limited to embeddings of Eulerian digraph...

In this paper we present some new sufficient conditions for equality of edge-connectivity and minimum degree of graphs and digraphs as well as of bipartite graphs and digraphs.

In this paper, we completely determine the connectivity of every infinite circulant digraphs and prove that almost all infinite circulant digraphs are infinitely strongly connected and therefore have both oneand two-way infinite Hamiltonian paths.

Given a digraph D, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We in...

In the first part of this paper we give necessary and sufficient conditions for some special classes of digraphs to have a (k, l)-kernel. One of them is the duplication of a set of vertices in a digraph. This duplication come into being as the generalization of the duplication of a vertex in a graph (see [4]). Another one is the D-join of a digraph D and a sequence α of nonempty pairwise disjoi...