a 2-emph{rainbow dominating function} (2rdf) on a graph $g=(v, e)$ is afunction $f$ from the vertex set $v$ to the set of all subsets of the set${1,2}$ such that for any vertex $vin v$ with $f(v)=emptyset$ thecondition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled. a 2rdf $f$ isindependent (i2rdf) if no two vertices assigned nonempty sets are adjacent.the emph{weight} of a 2rdf $f$ is the value $o...

Given a graph G = (V, E), the subdivision of an edge e = uv ∈ E(G) means the substitution of the edge e by a vertex x and the new edges ux and xv. The domination subdivision number of a graph G is the minimum number of edges of G which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of G...

We provide a simple constructive characterization for trees with equal domination and independent domination numbers, and for trees with equal domination and total domination numbers. We also consider a general framework for constructive characterizations for other equality problems.

For a simple graph G, the independent domination number i(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish upper bounds for the independent domination number of K1,k+1-free graphs, as functions of the order, size and k. Also we present a lower bound for the size of connected graphs with given order and value of independent domination ...

The paired bondage number (total restrained bondage number, independent bondage number, k-rainbow bondage number) of a graph G, is the minimum number of edges whose removal from G results in a graph with larger paired domination number (respectively, total restrained domination number, independent domination number, k-rainbow domination number). In this paper we show that the decision problems ...

We show that the diameter of a total domination vertex-critical graph is at most 5(γt −1)/3, and that the diameter of an independent domination vertex-critical graph is at most 2(i− 1). For all values of γt ≡ 2 (mod 3) there exists a total domination vertex-critical graph with the maximum possible diameter. For all values of i ≥ 2 there exists an independent domination vertex-critical graph wit...

A set S V is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. A set S V is an independent set of vertices if no two vertices in S are adjacent. The independence number, B0 (G), is the maximum cardinalit...

In this article we study some variants of the domination concept attending to the connectivity of the subgraph generated by the dominant set. This study is restricted to maximal outerplanar graphs. We establish tight combinatorial bounds for connected domination, semitotal domination, independent domination and weakly connected domination for any n-vertex maximal outerplaner graph.