On trees with equal domination and total outer-independent domination numbers (Q2811823)

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a dominating set if every vertex in \(V\backslash S\) is adjacent to at least one vertex in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set in \(G\). The set \(S\) is a total outer-independent dominating set if every vertex of \(G\) has a neighbor in \(S\), and the set \(V\backslash S\) is independent. The total outer-independent domination number of \(G\) is denoted by \(\gamma_t^{oi}(G)\) and is the minimum number of a total outer-independent dominating set of \(G\). The characterization of graphs with equal domination and total outer-independent domination number is an interesting problem in domination theory. The author of this paper characterizes all trees with this property. He introduces a family \(\mathcal{T}\) of trees \(T=T_k\) obtained recursively from \(T_{k-1}\) (note that \(T_1=P_4\)) by one of the seven operations stated in this paper. The main theorem of this paper states that ``The tree \(T\in \mathcal{T}\) if and only if \(\gamma(T)=\gamma_t^{oi}(T)"\).