A set S ⊆ V of vertices in a graph G = (V,E) is called a dominating set if every vertex in V − S is adjacent to at least one vertex in S. Domination in graphs is a well-studied branch of graph theory, and is the subject of two books by Haynes, Hedetniemi and Slater [8, 9]. However, about 90% of the papers on domination have considered only undirected graphs. Thus, relatively little is known abo...

For two or more classes of points in Rd with d ≥ 1, the class cover catch digraphs (CCCDs) can be constructed using the relative positions of the points from one class with respect to the points from one or all of the other classes. The CCCDs were introduced by Priebe et al. [C.E. Priebe, J.G. DeVinney, D.J. Marchette, On the distribution of the domination number of random class catch cover dig...

A subset $S$ of vertices a digraph $D$ is double dominating set (total $2$-dominating set) if every vertex not in adjacent from at least two $S$, and one (the subdigraph induced by has no isolated vertices). The domination number $2$-domination number) the minimum cardinality $D$. In this work, we investigate these concepts which can be considered as extensions graphs to digraphs, along with $2...

A domination graph of a digraph D, domeD), is created using the vertex set of D, V(D). There is an edge uv in domeD) whenever (u, z) or (v, z) is in the arc set of D, A(D), for every other vertex z E V(D). For only some digraphs D has the structure of domeD) been characterized. Examples of this are tournaments and regular digraphs. The authors have characterizations for the structure of digraph...

Let H = (V (H), A(H)) be a digraph possibly with loops and D = (V (D), A(D)) a digraph whose arcs are colored with the vertices of H (this is what we call an H-colored digraph); i.e. there exists a function c : A(D) → V (H); for an arc of D, f = (u, v) ∈ A(D), we call c(f) = c(u, v) the color of f . A directed walk (directed path) P = (u0, u1, . . . , un) in D will be called an H-walk (H-path) ...

Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A1, . . . , At of independent vertices. A set U = ∪i∈SAi is called a dominating set of size |S| if for any vertex v ∈ ∪i/ ∈SAi there is a w ∈ U such that (w, v) ∈ E(D). Let β(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main r...

We study the complexity of two dual covering and packing distance-based problems Broadcast Domination Multipacking in digraphs. A dominating broadcast a digraph D is function $$f:V(D)\rightarrow {\mathbb {N}}$$ such that for each vertex v D, there exists t with $$f(t)>0$$ having directed path to length at most f(t). The cost f sum f(v) over all vertices v. multipacking set S every integer d, ar...