We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for infinite families, exact values are established; in all other the lower and upper bounds with small gaps given. also define singleton rainbow domination, where sets assigned have a cardinality of, at most, one, provide analogous this special case domination.

An edge (vertex) coloured graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph G is the smallest number of colours needed for a rainbow edge (vertex) colouring of G. In this paper we propose a very simple approach to studying rainbow connectivity in...

A set of vertices S is said to dominate the graph G if for each v / ∈ S, there is a vertex u ∈ S with u adjacent to v. The minimum cardinality of any dominating set is called the domination number of the graph G and is denoted by γ(G). A dominating set D of a graph G = (V,E) is a nonsplit dominating set if the induced graph 〈V − D〉 is connected. The nonsplit domination number γns(G) of the grap...

For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2...

The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt(G) of a graph G and we show that for any connected graph G of order at least two, msdγt(G) ≤ 3. We show that...

A graph G is domination dot-critical, or just dot-critical, if contracting any edge decreases the domination number. It is totally dot-critical if identifying any twovertices decreases the domination number. In this paper, we study an open question concerning of the diameter of a domination dot-critical graph G. © 2008 Elsevier B.V. All rights reserved.

In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of Kn, there is a rainbow path on (3/4− o(1))n vertices, improving on the previously best bound of (2n + 1)/3 from [?]. Similarly, a k-rainbow path in a proper...

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 graph G is 3-domination critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. It was proved by Favaron et al. that α ≤ δ + 2 for any connected 3-domination critical graph. Denote by τ (G) the toughness of a graph G . Recently Chen et al. conjectured that a connected 3-domination critical graph G is Hamilton-connected if and only if τ (G) > 1 and showed the co...

In this paper, we consider the problem of reducing semitotal domination number a given graph by contracting k edges, for some fixed k≥1. We show that can always be done with at most 3 edge contractions and further characterise those graphs requiring 1, 2 or contractions, respectively, to decrease their number. then study complexity k=1 obtain in particular complete dichotomy monogenic classes.