Let G = (V,A) be a directed graph without parallel arcs, and let S ⊆ V be a set of vertices. Let the sequence S = S0 ⊆ S1 ⊆ S2 ⊆ · · · be defined as follows: S1 is obtained from S0 by adding all out-neighbors of vertices in S0. For k > 2, Sk is obtained from Sk−1 by adding all vertices w such that for some vertex v ∈ Sk−1, w is the unique out-neighbor of v in V \ Sk−1. We set M(S) = S0 ∪S1 ∪ · ...

We develop an O(n 3) algorithm for deciding if an n-vertex digraph has a subset of vertices with the property that each vertex of the graph has an even number of arcs into the subset. This algorithm allows us to give a combinatorial interpretation of Gauss-Jordan and Gauss elimination on square boolean matrices. In addition to solving this independence-mod-2 (even) set existence problem we also...

Recently, Hedetniemi et al. introduced (1, 2)-domination in graphs, and the authors extended that concept to (1, 2)-domination graphs of digraphs. Given vertices x and y in a digraph D, x and y form a (1, 2)-dominating pair if and only if for every other vertex z in D, z is one step away from x or y and at most two steps away from the other. The (1, 2)-dominating graph of D, dom1,2 (D) , is def...

Let D = (V,A) be a finite and simple digraph. A II-rainbow dominating function (2RDF) of a digraph D is a function f from the vertex set V to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V with f(v) = ∅ the condition ⋃ u∈N−(v) f(u) = {1, 2} is fulfilled, where N−(v) is the set of in-neighbors of v. The weight of a 2RDF f is the value ω(f) = ∑ v∈V |f(v)|. The 2-rainbow d...

We use the domination number of a parametrized random digraph family called proportional-edge proximity catch digraphs (PCDs) for testing multivariate spatial point patterns. This digraph family is based on relative positions of data points from various classes. We extend the results on the distribution of the domination number of proportional-edge PCDs, and use the domination number as a stati...

This paper introduces a notion of a random threshold directed graph, extending the work of Reilly and Scheinerman in the undirected case and closely related to random Ferrers digraphs. We begin by presenting the main definition: D is a threshold digraph provided we can find a pair of weighting functions f, g : V (D)→ R such that for distinct v, w ∈ V (D) we have v → w iff f(v) + g(w) > 1. We al...

Unlike graphs, digraphs or binary relational structures, the 2-structures do not define precise links between vertices. They only yield an equivalence between links. Also 2-structures provide a suitable generalization in the framework of clan decomposition. Let σ be a 2-structure. A subset C of V (σ) is a clan of σ if for each v ∈ V (σ) ∖ C, v is linked in the same way to all the elements of C....