On The Partitions Of The Dominating Set In Graphs

Domination theory of graphs more and more becomes the focus of people’s research andattention with the development of graph theory. The problem of graph’s dominating setpartition is studied in this paper which includes vertices dominating set partition and edgedominating set partition.First, vertex or edge (total) dimination number of some graphs is introduced. These lay asolid foundation for the problem of graph’s (total) dominating set partition.Then, dominating set partition, total dominating set partition and antidomatic partitionare introduced. Through seriously study, we obtained the bounds of (total) domatic number,and determined exact (total) domatic number for some special graphs.Next, edge dominating set partition are studied in three aspects similarly: edgedominating set partition, total edge dominating set partition and edge antidomatic partition.The edge domatic number of the generalized fan graph F (m, n)and the total edge domaticnumber of fan graphFn are determined on the basis of existing results. Edge antidomaticnumber of graphs are defined on the definition of antidomatic number. And the bounds ofgraphs’ edge antidomatic number and some special graphs’ exactly edge antidomatic numberare studied.Finally, this paper’s main results are summarized briefly and the further research worksare introduced.