Global color class domination partition of a graph

Abstract

Color class domination partition was suggested by E. Sampathkumar and it was studied in [1]. A proper color partition of a finite, simple graph G is called a color class domination partition (or cd-partition) if every color class is dominated by a vertex. This concept is different from dominator color partition introduced in [[2], [3]] where every vertex dominates a color class. Suppose G has no full degree vertex (that is, a vertex which is adjacent with every other vertex of the graph). Then a color class may be independent from a vertex outside the class. This leads to Global Color Class Domination Partition. A proper color partition of G is called a Global Color Class Domination Partition if every color class is dominated by a vertex and each color class is independent of a vertex outside the class. The minimum cardinality of a Global Color Class Domination Partition is called the Global Color Class Domination Partition Number of G and is denoted by Xgcd(G). In this paper a study of this new parameter is initiated and its relationships with other parameters are investigated.