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Given a digraph D = (V, A), a set B ⊂ V is a packing set in D if there are no arcs joining vertices of B and for any two vertices x, y ∈ B the sets of in-neighbors of x and y are disjoint. The set S is a dominating set (an open dominating set) in D if every vertex not in S (in V ) has an in-neighbor in S. Moreover, a dominating set S is called a total dominating set if the subgraph induced by S has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We prove that the two optimal sets concerning packing and domination achieve the same value for directed trees, and give some applications of it. We also show analogous equalities for all connected contrafunctional digraphs, and characterize all such digraphs D for which such equalities are satisfied. Moreover, sharp bounds on the maximum and the minimum cardinalities of packing and dominating sets, respectively, are given for digraphs. Finally, we present solutions for two open problems, concerning total and open dominating sets of minimum cardinality, pointed out in [Australas. J. Combin. 39 (2007), 283-292].

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We define a k-total limited packing number in a graph, which generalizes the concept of open packing number in graphs, and give several bounds on it. These bounds involve many well known parameters of graphs. Also, we establish a connection among the concepts of tuple domination, tuple total domination and total limited packing that implies some results.

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For a graph $G=(V(G),E(G))$ , an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$ , either $v$ is adjacent to a vertex assigned $2$ under $f$ or $v$ is adjacent to least two vertices assigned $1$ under $f$ . The weight of an ID function is $\sum_{v\in V(G)}f(v)$ . The Italian domination number is the minimum weight taken over all ID functions of $G$ . In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function) $f$ of $G$ is an ID function of $G$ for which the subgraph induced by $\{v\in V(G)\mid f(v)=0\}$ has no isolated vertices , and the restrained Italian domination number $\gamma_{rI}(G)$ is the minimum weight taken over all RID functions of $G$ . We first prove that the problem of computing this parameter is NP-hard, even when restricted to bipartite graphs and chordal graphs as well as planar graphs with maximum degree five. We prove that $\gamma_{rI}(T)$ for a tree $T$ of order $n\geq3$ different from the double star $S_{2,2}$ can be bounded from below by $(n+3)/2$ . Moreover, all extremal trees for this lower bound are characterized in this paper. We also give some sharp bounds on this parameter for general graphs and give the characterizations of graphs $G$ with small or large $\gamma_{rI}(G)$ .

An outer independent (double) Roman dominating function is a (double) Roman dominating function f for which the set of vertices assigned 0 under f is independent. The outer independent (double) Roman domination number (γ oid R (G)) γ oi R (G) is the minimum weight taken over all outer independent (double) Roman dominating functions of G. A vertex cover number β(G) is the minimum size of any vertex cover sets of a graph G. In this work, we present some contributions to the study of outer independent double Roman domination in graphs. Characterizations of the families of all connected graphs with small outer independent double Roman domination numbers, and tight lower and upper bounds on this parameter are given. We also prove that the decision problem associated with γ oid R (G) is NP-complete even when restricted to planar graphs with maximum degree at most four. We moreover prove that 2β(T ) + 1 ≤ γ oid R (T ) ≤ 3β(T ) for any tree T , and show that each integer between the lower and upper bounds is realizable. Finally, we give an exact formula for this parameter concerning the corona graphs. Keywords (Outer independent) double Roman domination number • Roman domination number • Vertex cover number Mathematics Subject Classification 05C69Communicated by Behruz Tayfeh-Rezaie.

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turán [8], we present a sharp lower bound on Kr+1-free graphs for r ≥ 2. Applying the concept of total limited packing we bound the signed total domination number of G with δ(G) ≥ 3 from above by n − 2 2ρo(G)+δ−3 2 . Also, we prove that γst(T ) ≤ n − 2(s − s ) for any tree T of order n, with s support vertices and s support vertices of degree two. Moreover, we characterize all trees attaining this bound.