Search

A vertex subset S of a graph G = (V , E) is a [ 1 , 2 ] -dominating set if each vertex of V ∖ S is adjacent to either one or two vertices in S. The minimum cardinality of a [ 1 , 2 ] -dominating set of G , denoted by γ [ 1 , 2 ] (G) , is called the [ 1 , 2 ] -domination number of G. In this paper, the [ 1 , 2 ] -domination and the [ 1 , 2 ] -total domination numbers of the generalized Petersen graphs P (n , 2) are determined. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in four proper vertex colorings of graphs, with additional domination property. In the dominator colorings , strong colorings and strict strong colorings of a graph G , every vertex has to dominate at least one color class. Conversely, in the dominated colorings of G , every color class has to be dominated by at least one vertex. We study arbitrary graphs as well as P 4 -sparse graphs, P 5 -free graphs, bounded treewidth graphs and claw-free graphs. [ABSTRACT FROM AUTHOR]