Let D be a digraph. V (D) denotes the set of vertices of D; a set N ⊆ V (D) is said to be a k-kernel of D if it satisfies the following two conditions: for every pair of different vertices u, v ∈ N it holds that every directed path between them has length at least k and for every vertex x ∈ V (D) − N there is a vertex y ∈ N such that there is an xy-directed path of length at most k − 1. In this...

Bang-Jensen and Hell conjectured in 1990 (using the language of graph homomorphisms) a CSP dichotomy for digraphs with no sources or sinks. The conjecture states that the constraint satisfaction problem for such a digraph is tractable if each component of its core is a circle and is NP-complete otherwise. In this paper we prove this conjecture, and, as a consequence, a conjecture of Bang-Jensen...

This paper presents additional conditions that are needed to create proper digraphs representation of the characteristic polynomial. Contrary to currently used methods (like canonical forms) digraphs representations allow to find a complete set of all possible realisations instead of only a few realisations. In addition, all realisations in the set are minimal. Proposed additional conditions on...

Undirected graphs of pathwidth at most one are characterized by two forbidden minors i.e., (i) K3 the complete graph on three vertices and (ii) S2,2,2 the spider graph with three legs of length two each [BFKL87]. Directed pathwidth is a natural generalization of pathwidth to digraphs. In this paper, we prove that digraphs of directed pathwidth at most one are characterized by a finite number of...

Digraphs in which ordered pairs of outand in-degrees of vertices are mutually distinct are called irregular, see Gargano et al. [3]. Our investigations focus on the problem: what are possible sizes of irregular digraphs (oriented graphs) for a given order n? We show that those sizes in both cases make up integer intervals. The extremal sizes (the 612 Z. Dziechcińska-Halamoda, Z. Majcher, ... en...

An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...