A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the notion of k-dominating ...

An outer-connected dominating set for an arbitrary graph G is a set D̃ ⊆ V such that D̃ is a dominating set and the induced subgraph G[V \ D̃] be connected. In this paper, we focus on the outerconnected domination number of the product of graphs. We investigate the existence of outer-connected dominating set in lexicographic product and Corona of two arbitrary graphs, and we present upper bounds f...

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by γc(G) + 2, where γc(G) is th...

Let k ∈ N and let G be a graph. A function f : V (G) → 2 is a rainbow function if, for every vertex x with f(x) = ∅, f(N(x)) = [k]. The rainbow domination number γkr(G) is the minimum of ∑ x∈V (G) |f(x)| over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs.

Abstract By suitably adjusting the tropical algebra technique we compute rainbow independent domination numbers of several infinite families graphs including Cartesian products $$C_n \Box P_m$$ C n ? P m </mm...

In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

The dominating set problem is a core NP-hard problem in combinatorial optimization and graph theory, and has many important applications. Baker [JACM 41,1994] introduces a k-outer planar graph decomposition-based framework for designing polynomial time approximation scheme (PTAS) for a class of NP-hard problems in planar graphs. It is mentioned that the framework can be applied to obtain an O(2...