Let D = (V , A) be a directed graph (digraph) without loops nor multiple arcs. A set of vertices S of a digraph D is a (k, l)-kernel of D if and only if for any two vertices u, v in S, d(u, v) ≥ k and for any vertex u in V \ S there exists v in S such that d(u, v) ≤ l. A digraph D is called quasi-transitive if and only if for any distinct vertices u, v, w of D such that u→ v → w, then u and w a...

For digraphs D and H , a mapping f : V (D) → V (H) is a homomorphism of D to H if uv ∈ A(D) implies f (u)f (v) ∈ A(H). For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H). An optimization version of the homomorphism problemwasmotivated by a real-world problem in defence logistics and was introduced ...

Let Zd be the additive group of 1×n row vectors over Zd. For an n×nmatrix T over Zd and ω ∈ Z n d , the affine transformation FT,ω of Z n d sends x to xT +ω. Let 〈α〉 be the cyclic group generated by a vector α ∈ Zd . The affine transformation coset pseudo-digraph TCP (Zd , α, FT,ω) has the set of cosets of 〈α〉 in Z n d as vertices and there are c arcs from x+ 〈α〉 to y+ 〈α〉 if and only if the nu...

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...