TBSs in some minimum coverings

Let ( X , B ) be a ( λ K n , G ) -covering with excess E and a blocking set T . Let Γ 1 , Γ 2 , … , Γ s be all connected components of E with at least two vertices (note that s = 0 if E = 0̸ ). The blocking set T is called tight if further V ( Γ i ) ∩ T ≠ 0̸ and V ( Γ i ) ∩ ( X ∖ T ) ≠ 0̸ for 1 ≤ i ≤ s . In this paper, we give a complete solution for the existence of a minimum ( λ K n , G ) -covering admitting a blocking set (BS), or a tight blocking set (TBS) for any λ and when G = K 3 and G = K 3 + e .