Inequalities between the Kk-isolation number and the independent Kk-isolation number of a graph

Abstract For a graph G , a vertex subset S ⊆ V ( G ) is said to be K k -isolating if G − N G [ S ] does not contain a clique K k as a subgraph. The K k -isolation number of G , denoted by ι k ( G ) , is the minimum cardinality of a K k -isolating set of G . The set S is said to be independent K k -isolating if S is a K k -isolating set of G and G [ S ] has no edge. The independent K k -isolation number of G , denoted by ι k ′ ( G ) , is the minimum cardinality of an independent K k -isolating set of G . A K 1 -isolating set (independent K 1 -isolating set) is clearly a dominating set (independent dominating set). Hence ι 1 ( G ) = γ ( G ) , the domination number of G , and ι 1 ′ ( G ) = i ( G ) , the independent domination number of G . For any graph G of maximum degree Δ ( G ) and any integer k ≥ 1 , we prove that ι k ′ ( G ) ∕ ι k ( G ) ≤ Δ ( G ) − 2 Δ ( G ) + 2 and that ι k ′ ( G ) ≤ ( r − 2 ) ( ι k ( G ) − 1 ) + 1 if G is K 1 , r -free. This respectively generalizes results from Furuya et al. (2014) and from Bollobas and Cockayne (1979) established for k = 1 .