Abstract

For a connected graph G = (V,E), a hull set S in a connected graph G is called a hull dominating set of G if S is both hull set and a dominating set of G. The hull domination number γ_h (G) of 𝐺 is the minimum cardinality of a hull dominating set of G. A hull dominating set Sof G is called a minimal hull dominating set if there is no proper subset S of S such that S is a hull dominating set of G. The upper hull domination number γ_h^+ (G)=m ax⁡{|S|:S is a minimal hull dominating set of G}. Some general properties satisfied by this concept are studied. Connected graphs of order p≥2 with upper hull domination number p or p-1 are characterized. It is shown that for every positive integera≥2, there exists a connected graph G such that γ_h (G)= a and γ_h^+ (G)= 2a.