Some Signed Domination Numbers Of Graphs

Dominating parameter on graph was introduced by Ore and Berge. Cockayne and Hedetnieni gave a survey on the results of dominating parameter on graphs,and denoted byγthe domination number of a graph G. The dominating parameter of graphs has extensively been investigated in the past.Haynes et al.systemically summarized the dominating parameter.The study of dominating parameter in graphs has been an important branch in graph theory and combinatorics.Some interesting results on signed domination number for a few special graphs have been obtained in recent years.However,there are many open problems on the signed domination number of general graphs.Based on some known results,in this thesis we study the problems of signed domination numbers.The main results obtained in this thesis are as follows:(1) Two tight lower bounds on the domination number of general graphs are established by a constructive method.(2) Determine the signed total domination numbers for some special graphs such as complete graphs,stars,fans,wheels,complete multi-part graphs,etc.(3) Define q-signed domination number,obtain q-signed domination number of a graph G with respect to different parameters,and prove that one of these results is sharp. (4) Give a tight upper bound on upper q-signed domination number of k-regular graphs.