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Excellent trees. (English) Zbl 0995.05036

Let \(\mu(G)\) be a numerical invariant of a graph \(G\) defined in such a way that it is the minimum or maximum number of vertices of a set \(S\subseteq V(G)\) with a given property \(P\). A set with the property \(P\) and with \(\mu(G)\) vertices in \(G\) is called a \(\mu(G)\)-set. If a vertex of \(G\) is contained in some \(\mu(G)\)-set, it is called \(\mu\)-good and otherwise it is called \(\mu\)-bad. If every vertex of \(G\) is \(\mu\)-good, the graph \(G\) is called \(\mu\)-excellent. The property of being a \(\mu\)-excellent tree is studied for three invariants \(\mu(G)\), namely the domination number \(\gamma(G)\), the irredundance number \(\text{ir}(G)\) and the independence number \(\beta(G)\).

MSC:

05C05 Trees
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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