On Tree Partitions

A splitting partition for a graph \(G = (V, E)\) is a partition of \(V\) into sets \(R\), \(B\), and \(U\) so that the subgraphs induced by \(V – R\) and \(V – B\) are isomorphic. The splitting number \(\mu(G)\) is the size of \(|R|\) for any splitting partition which maximizes \(|R|\). This paper determines \(\mu(G)\) for trees of maximum degree at most three and exactly one degree two vertex and for trees all of whose vertices have degree three or one.