A dicut in a directed graph is cut for which all of its edges are to common side the cut. famous theorem Lucchesi and Younger states that every finite digraph least size an edge set meeting equals maximum number disjoint dicuts digraph. In this first paper out series two papers, we conjecture version using more structural description min-max property infinite digraphs. We show can be reduced co...

The imbalance of a vertex v in a digraph D is defined as a(v) = d + (v)−d − (v), where d + (v) and d − (v) respectively denote the outdegree and indegree of vertex v. The imbalance sequence of D is formed by listing vertex imbalances in nondecreasing order. We define a minimally cyclic digraph as a connected digraph which is either acyclic or has exactly one oriented cycle whose removal disconn...

A time-stamped graph is an undirected graph with a real number on each edge. Vertex u in#uences vertex v if there is a non-decreasing path from u to v. The associated in#uence digraph of a time-stamped graph is the directed graph that records the in#uences. Among other results, we determine for what n and t there exists a time-stamped graph whose associated in#uence digraph has n vertices and t...

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann (1998) raised the following question: Let D be a strong semicomplete multipartite digraph with a longest path of length l. Does there exist a strong spanning oriented subgraph of D with a lo...

A consecutive-d digraph is a digraph G(d, n, q, T) whose n nodes are labeled by the residues modulo n and a link from node i to node j exists if and only if j I pi + k (mod n) for some k with r 5 k 5 r + d 1. Consecutive-d digraphs are used as models for many computer networks and multiprocessor systems, in which the existence of a Hamiltonian circuit is important. Conditions for a consecutive-...

Let G = (V,E) be an undirected graph and let π = {V1, V2, . . . , Vk} be a partition of the vertices V of G into k blocks Vi. From this partition one can construct the following digraph D(π) = (π,E(π)), the vertices of which correspond one-to-one with the k blocks Vi of π, and there is an arc from Vi to Vj if every vertex in Vj is adjacent to at least one vertex in Vi, that is, Vi dominates Vj ...

A digraph D is primitive if, for some positive integer r, there is a u ! v walk of length r for every pair u; v of vertices of D. The minimum such r is called the exponent of D, denoted exp(D). We prove that if a primitive digraph D of order n is an Abelian Cayley digraph with degree k 4, then exp(D) max n 5; n dk=2e 1 o : If we assume only that the primitive digraph D is vertex-transitive, the...

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy Problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynom...

Let D be a digraph. Let k be a natural number with k 2. A set J ⊆ V (D) will be called a k-kernel of the digraph D iff: 1) For each x, x′ ∈ J , x = x′ we have dD(x, x′) k and 2) For each y ∈ V (D) − J , there exists x ∈ J such that dD(y, x) ≤ k − 1. In [4], for any digraph D we constructed a digraph s(S) such that D has a k-kernel iff s(S) has a k-kernel. In this paper we prove that the number ...