A subset S of V is called a dominating set in G if every vertex in V-S is adjacent to at least one vertex in S. A Dominating set is said to be Fuzzy Double Dominating set if every vertex in V-S is adjacent to at least two vertices in S. The minimum cardinality taken over all, the minimal double dominating set is called Fuzzy Double Domination Number and is denoted by γ fdd (G). The minimum numb...

compiled April 30, 2009 from draft version hg:e0660c153c0b:79 An acyclic coloring of a graph is a proper vertex coloring such that the subgraph induced by the union of any two color classes is a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. The acyclic and star chromatic numbers ...

For a graph G and integer k ≥ 1, we define the token graph Fk(G) to be the graph with vertex set all k-subsets of V (G), where two vertices are adjacent in Fk(G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of Fk(G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever...

The oriented chromatic number of a graph G is the maximum, taken over all orientations of G, of the minimum number of colours in a proper vertex colouring of G such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, whose average degree is at least logarithmic in the number of vertices. Fo...

A vertex (edge) coloring φ : V → {1, 2, . . . , t} (φ′ : E → {1, 2, . . . , t}) of a graph G = (V,E) is a vertex (edge) t-ranking if, for any two vertices (edges) of the same color, every path between them contains a vertex (edge) of larger color. The vertex ranking number χr(G) (edge ranking number χ′ r (G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we...

Effective solutions to problems encountered in networks are often based on whether the elementary set can be partitioned into classes according to some specific criteria. The chromatic number of a graph G(V,E) is the minimum number of colours needed to colour the vertices of G such that no two adjacent vertices have the same colour. An independent set D of G is called an efficient dominating se...