Partitioning of Trees Having Maximum Degree at Most Three

The vertices \(V\) of trees with maximum degree three and \(t\) degree two vertices are partitioned into sets \(R\), \(B\), and \(U\) such that the induced subgraphs \(\langle V – R \rangle\) and \(\langle V – B \rangle\) are isomorphic and \(|U|\) is minimum. It is shown for \(t \geq 2\) that there is such a partition for which \(|U| = 0\) if \(t\) is even and \(|U| = 1\) if \(t\) is odd. This extends earlier work by the authors which answered this problem when \(t = 0\) or \(1\).