Total restrained domination in trees (Q879341)

A subset \(S\) of the vertex set \(V\) of a graph \(G(V, E)\) is a total restrained dominating set if every vertex is adjacent to a vertex in \(S\) and every vertex of \(V-S\) is adjacent to a vertex in \(V-S\). The total restrained domination number of \(G\) is the smallest cardinality of a total restrained dominating set of \(G\). The paper gives lower bounds for the total restrained domination number for trees of order \(n\) and of order \(n = 0\) mod 4. It also constructively characterizes the extremal trees of order \(n\) achieving these lower bounds.