A digraph G is a difference digraph iff there exists an S ⊂ IN such that G is isomorphic to the digraph DD(S) = (V,A), where V = S and A = {(i, j) : i, j ∈ V ∧ i− j ∈ V }. For some classes of digraphs, e.g. alternating trees, oriented cycles, tournaments etc., it is known, under which conditions these digraphs are difference digraphs (cf. [5]). We generalize the so-called sourcejoin (a construc...

In this paper we study the problem of existence of a crossing-free acyclic hamiltonian path completion (for short, HP-completion) set for embedded upward planar digraphs. In the context of book embeddings, this question becomes: given an embedded upward planar digraph G, determine whether there exists an upward 2-page book embedding of G preserving the given planar embedding. Given an embedded ...

A k-digraph is a digraph in which every vertex has outdegree at most k. A (k ∨ l)digraph is a digraph in which a vertex has either outdegree at most k or indegree at most l. Motivated by function theory, we study the maximum value Φ(k) (resp. Φ(k, l)) of the arc-chromatic number over the k-digraphs (resp. (k ∨ l)-digraphs). El-Sahili [3] showed that Φ(k, k) ≤ 2k+1. After giving a simple proof o...

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...