Partial Domination of Network Models *

Partial domination [8] is a theory first proposed in 2015 on the basis of domination theory, which has a wide range of research value. Let G = (V, E) be a graph and F be a family of graphs, a subset S ⊆ V is called an F-isolating set of G if G[V \NG[S]] does not contain F as a subgraph for all F ∈ F. If F = {K2}, S is an isolating set of G if G[V \NG[S]] does not contain K2. The isolation number of G is the minimum cardinality of an isolating set of G, denoted by ι(G). The hypercube network and n-star network are the basic models for interconnection networks, and they have many attractive topological properties. In this paper, we investigate the sharp bounds of the isolation numbers of the hypercube network Qn and n-star network Sn, and obtain (2n−1)\n ≤ ι(Qn) ≤ 2n−3 for any positive integer n ≥ 4 and (n·(n2−2)!)\2 ≤ ι(Sn) ≤ (n − 1)! for any positive integer n ≥ 2. AMS subject classification: 05C05, 05C12, 05C76