Domination in digraphs and their direct and Cartesian products

Abstract

A dominating (respectively, total dominating) set S of a digraph D is a set of vertices in D such that the union of the closed (respectively, open) out-neighborhoods of vertices in S equals the vertex set of D. The minimum 1 2 γ H ( ( ) + 1). This inequality is sharp as demonstrated by an infinite family of examples. Ditrees T and

Orientable domination in product-like graphs

“…The classical result of Meir and Moon [13] states that the domination number of a tree T is equal to the 2-packing number of T , which is defined as the maximum number of pairwise disjoint closed neighborhoods in T . In [3] this result was extended to the context of digraphs. (See also Mojdeh, Samadi and G. Yero [14,Theorem 5] where the special case of this result was proved for orientations of trees.)…”

“…Let X = P 3 K 3 . Set V (P 3 ) = {a, x, y} with E(P 3 ) = {ax, ay}, and V (K 3 ) = [3]. Consider an arbitrary orientation X f of X.…”

“…The mentioned extension to digraphs [3, Theorem 1.2] asserts that if T is a digraph whose underlying graph is a tree, then ρ(T ) = γ(T ). The authors then asked whether the result can be extended to all acyclic digraphs; see [3,Problem 1]. Recall that a digraph is acyclic if it contains no directed cycles.…”

“…We also study the orientable domination number in specific classes of lexicographic product graphs. In particular, a specific orientation of the graph C 2k+1 •K s allows us to provide a negative answer to the problem from [3] asking whether the domination number of an acyclic digraph is equal to its packing number.…”

On the Packing Partitioning Problem on Directed Graphs

“…Thus, the presence of an open version of the concept of packing (packing partition) can be expected. Brešar et al [8] defined the open packing number in digraphs as follows. A subset We observe that the direct product D × F is empty (that is, A(D × F) = ∅) if at least one of the factors D and F is empty.…”