A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function f is the sum, u∈V (G) f (u), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in G. A total Roman domination function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The total Roman domination number is the minimum weight of a total Roman domination function on G. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.2010 Mathematics Subject Classification. 05C69.
An outer-independent Italian dominating function (OIIDF) on a graph G with vertex set V (G) is defined as a function f : V (G) → {0, 1, 2}, such that every vertex v ∈ V (G) with f (v) = 0 has at least two neighbors assigned 1 under f or one neighbor w with f (w) = 2, and the set {u ∈ V | f (u) = 0} is independent. The weight of an OIIDF f is the value w(f) = u∈V (G) f (u). The minimum weight of an OIIDF on a graph G is called the outer-independent Italian domination number γ oiI (G) of G. In this paper, we initiate the study of the outer-independent Italian domination number and present the bounds on the outer-independent Italian domination number in terms of the order, diameter, and vertex cover number. In addition, we establish the lower and upper bounds on γ oiI (T) when T is a tree and characterize all extremal trees constructively. We also give the Nordhaus-Gaddum-type inequalities. INDEX TERMS Outer-independent Italian domination, Italian domination, trees.
The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number. In this paper we present new upper bounds for b(G) in terms of girth, order and Euler characteristic.
A Roman dominating function (RD-function) on a graph G = (V (G), E(G)) is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The weight f (V (G)) of a RD-function f on G is the value Σ v∈V (G) f (v). The Roman domination number γ R (G) of G is the minimum weight of a RD-function on G. The six classes of graphs resulting from the changing or unchanging of the Roman domination number of a graph when a vertex is deleted, or an edge is deleted or added are considered. We consider relationships among the classes, which are illustrated in a Venn diagram. A graph G is Roman domination k-critical if the removal of any set of k vertices decreases the Roman domination number. Some initial properties of these graphs are studied. The γ R -graph of a graph G, is any graph which vertex set is the collection D R (G) of all minimum weight RDfunctions on G. We define adjacency between any two elements of D R (G) in several ways, and initiate the study of the obtained γ R -graphs.
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