A subset D of ( ) V G is called an equitable dominating set if for every ( ) v V G D   there exists a vertex u D  such that ( ) uv E G  and | ( ) ( ) | 1 deg u deg v   . A subset D of ( ) V G is called an equitable independent set if for any , u D v   ( ) e N u for all { } v D u   . The concept of equi independent equitable domination is a combination of these two important concepts. ...

A Roman domination function on a graph G is a function r : 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 function is the value r(V (G)) = ∑ u∈V (G) r(u). The Roman domination number γR(G) of G is the minimum weight of a Roman domination function on G . "Roman Criticality" has been ...

A dominating set D of G is called a split dominating set of G if the subgraph induced by the subset V (G) − D is disconnected. The cardinality of a minimum split dominating set is called the minimum split domination number of G. Such subset and such number was introduced in [4]. In [2], [3] the authors estimated the domination number of products of graphs. More precisely, they were study produc...

Several of the best known problems and conjectures in graph theory arise in studying the behavior of a graphical invariant on a graph product. Examples of this are Vizing’s conjecture, Hedetniemi’s conjecture and the calculation of the Shannon capacity of graphs, where the invariants are the domination number, the chromatic number and the independence number on the Cartesian, categorical and st...

We consider families of analytic functions with Taylor coefficients-polynomials in the parameter λ: fλ(z) = ∑∞ k=0 ak(λ)z k, ak ∈ C[λ]. Let R(λ) be the radius of convergence of fλ. The “Taylor domination” property for this family is the inequality of the following form: for certain fixed N and C and for each k ≥ N + 1 and λ, |ak(λ)|R(λ) ≤ C max i=0,...,N |ai(λ)|R(λ). Taylor domination property ...

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework...

Roman domination is a historically inspired variety of general domination such that every vertex is labeled with labels from {0, 1, 2}. Roman domination number is the smallest of the sums of labels fulfilling condition that every vertex, labeled 0, has a neighbor, labeled 2. Using algebraic approach we give O(C) time algorithm for computing Roman domination number of special classes of polygrap...

Broadcast domination was introduced by Erwin in 2002, and it is a variant of the standard dominating set problem, such that vertices can be assigned various domination powers. Broadcast domination assigns a power f(v) 0 to each vertex v of a given graph, such that every vertex of the graph is within distance f(v) from some vertex v having f(v) 1. The optimal broadcast domination problem seeks t...

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...