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 three-valued function f defined on the vertices of a graph G = ( V, E), f : V 4 {-I. 0. I }, is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every 1~ t V, ,f(N[o]) > 1, where N[c] consists of I: and every vertex adjacent to 1’. The weight of a minus dominating function is f(V) = c f(u), over all vertices L: t V. The m...

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) ...

For a graph G, a function f : V (G) ! f?1; 0; +1g is called a minus-domination function of G if the closed neighborhood of each vertex of G contains strictly more

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...