On k-independence in trees

For a graph G , a set D ⊆ V ( G ) is a dominating set of G if every vertex in V ( G ) ∖ D has at least one neighbor in D . For any positive integer k , a set S ⊆ V ( G ) is a k -independent set of G if G [ S ] has maximum degree less than k . The k -independence number of G , denoted by α k ( G ) , is the maximum cardinality of a k -independent set in G . A set I ⊆ V ( G ) is a k -independent dominating set of G if I is both k -independent and dominating. The k -independent domination number of G , denoted by i k ( G ) , is the minimum cardinality of a k -independent dominating set in G . It is clear that i 1 ( G ) = i ( G ) ≤ α ( G ) = α 1 ( G ) , where i ( G ) and α ( G ) are the usual independent domination number and independence number of G , respectively. Let T be a tree with n ≥ 2 vertices. Very recently, Zhang and Wu proved that i 2 ( T ) ≤ 2 3 α 2 ( T ) and the present authors proved that i 3 ( T ) ≤ 3 5 α 3 ( T ) , and both bounds are sharp. These results lead to the natural question of whether there is an analogous relationship between i k ( T ) and α k ( T ) for general k ≥ 4 . In this paper, we give an affirmative answer to this question. For each k ≥ 4 , we prove that i k ( T ) ≤ k 2 k − 1 α k ( T ) and characterize all extremal trees for which the equality holds.