Multiple factor NordhausGaddum type results for domination and total domination
A NordhausGaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we examine the sum and product of ?t(G1),?t(G2),,?t(Gk) ? t ( G 1 ) , ? t ( G 2 ) ,  , ? t ( G k ) and the sum of ?(G1),?(G2),,?(Gk) ? ( G 1 ) , ? ( G 2 ) ,  , ? ( G k ) where G1?G2???Gk=Kn G 1 ? G 2 ? ? ? G k = K n for positive integers n n and k k , ?(G) ? ( G ) is the domination number and ?t(G) ? t ( G ) is total domination number of a graph G G . We show that ? j = 1 k ? ( G j ) ? ( k ? 1 ) n + 1 with equality if and only if Gi=Kn G i = K n for some i?{1,,k} i ? { 1 ,  , k } . For n?7 n ? 7 , 3?k?n?2 3 ? k ? n ? 2 and ?(Gi)?1 ? ( G i ) ? 1 for each i?{1,2,,k} i ? { 1 , 2 ,  , k } , we show that ? j = 1 k ? t ( G j ) ? ( k ? 1 ) ( n + 1 ) .
nordhausgaddum domination total domination