New results on singleton rainbow domination numbers of generalized Petersen graphs P(ck,k) are given. Exact values established for some infinite families, and lower upper bounds with small gaps given in all other cases.

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...

After 2-crossing-critical graphs were characterized in 2016, their most general subfamily, large 3-connected graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for domination independence number.

A subset D of the vertex set of a graph G is a (k, p)-dominating set if every vertex v ∈ V (G) \ D is within distance k to at least p vertices in D. The parameter γk,p(G) denotes the minimum cardinality of a (k, p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that γk,p(G) ≤ p p+k n(G) for any graph G with δk(G) ≥ k + p − 1, where the latter means that every vertex ...

A subset S ⊆ V in a graph G = (V,E) is a total [1, 2]-set if, for every vertex v ∈ V , 1 ≤ |N(v) ∩ S| ≤ 2. The minimum cardinality of a total [1, 2]-set of G is called the total [1, 2]-domination number, denoted by γt[1,2](G). We establish two sharp upper bounds on the total [1,2]-domination number of a graph G in terms of its order and minimum degree, and characterize the corresponding extrema...

Let γ(G) and γ2,2(G) denote the domination number and (2, 2)domination number of a graph G, respectively. In this paper, for any nontrivial tree T , we show that 2(γ(T )+1) 3 ≤ γ2,2(T ) ≤ 2γ(T ). Moreover, we characterize all the trees achieving the equalities.

‎A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$‎. ‎The total domination number of a graph $G$‎, ‎denoted by $gamma_t(G)$‎, ‎is~the minimum cardinality of a total dominating set of $G$‎. ‎Chellali and Haynes [Total and paired-domination numbers of a tree, AKCE International ournal of Graphs and Combinatorics 1 (2004)‎, ‎6...

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...