Lower bound on the domination number of a tree

Abstract

We prove that the domination number γ(T) of a tree T on n ≥ 3 vertices and with n 1 endvertices satisfies inequality γ(T) ≥ n+2−n 1 3 and we characterize the extremal graphs.

Smallest domination number and largest independence number of graphs and forests with given degree sequence

“…Improving a lower bound on the domination number of a tree T due to Lemanska , Desormeaux et al. showed a lower bound on , which implies for every tree T of order at least three.…”

Total restrained domination numbers of trees

“…Lemańska [6] has given a lower bound on the domination number of a tree T in terms of n(T ) and n 1 (T ).…”

“…Now we prove that if T is a tree of order at least 3 and 3 tr (T ) = n(T ) + 2 + 2n 1 (T ), then T belongs to the family R. To this aim we shall need the following results given in [6].…”

Lower bounds for the domination number

“…Lemanska shows in [20] that equality holds in Theorem 6 if and only if T is a tree such that the distance between any two leaves is congruent to 2 modulo 3. Since for trees, the number of cut-vertices is exactly n − l, equality holding in Theorem 6 is a sufficient condition for equality holding in the above theorem.…”