Strong equality of upper domination and independence in trees (Q2725015)

A subset \(S\) of the vertex set \(V(G)\) of a graph \(G\) is called dominating in \(G\), if each vertex \(v\in V(G)\) either is in \(S\), or is adjacent to a vertex of \(S\). A dominating set \(S\) in \(G\) is called minimal, if none of its proper subsets is dominating in \(G\). The maximum number of vertices of a minimal dominating set in \(G\) is the upper domination number \(\Gamma(G)\) of \(G\). A subset \(S\) of \(V(G)\) is called independent in \(G\), if no two vertices of \(S\) are adjacent in \(G\). The maximum number of vertices of an independent set in \(G\) is the independent number \(\beta(G)\) of \(G\). \(\Gamma(G)\) is said to equal strongly \(\beta(G)\), written \(\Gamma(G)\equiv \beta(G)\), if \(\Gamma(G)= \beta(G)\) and each minimal dominating set in \(G\) with \(\Gamma(G)\) vertices is also an independent set in \(G\). The paper gives a characterization of the trees \(T\) with \(\Gamma(T)\equiv \beta(T)\).