Total restrained domination in trees Original Research Article

Let image be a graph. A set image is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of image is adjacent to a vertex in image. The total restrained domination number of G, denoted by image, is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then image. Moreover, we show that if T is a tree of order image, then image. We then constructively characterize the extremal trees T of order n achieving these lower bounds.