Paired Domination in Trees

Abstract A set S of vertices in a graph G is paired dominating if every vertex adjacent to and the subgraph induced by contains perfect matching (not necessarily as an subgraph). The domination number, $$\gamma _{\mathrm{pr}}(G)$$ γ pr ( G ) , minimum cardinality . In this paper, we show that T tree order at least 2, then _{\mathrm{pr}}(T) \le 2\alpha (T) - \varphi (T)$$ T ≤ 2 α - φ where $$\alpha independence number $$\varphi $$P_3$$ P 3 -packing number. We present tight upper bound on terms its maximum degree $$\varDelta$$ Δ For $$\varDelta \ge 1$$ ≥ 1 n with \left( \frac{5\varDelta -4}{8\varDelta -4} \right) + \frac{1}{2}n_1(T) \frac{1}{4}n_2(T) \frac{\varDelta -2}{4\varDelta -2} \right)$$ 5 4 8 n + $$n_1(T)$$ $$n_2(T)$$ denote 1 respectively, Further, for all 3$$ As consequence result, $$n 2$$ \frac{5}{8} \frac{1}{4}n_2(T)$$ asymptotically best possible.