On the maximum size of resistant subgraphs in graphs with given average threshold

Let (G, tau) be a graph together with a threshold assignment tau : V (G) -> N such that 1 <= tau(v) <= d(v), where d(v) is the degree of v in G. A proper subset M subset of V(G) is called tau-resistant if for any vertex v is an element of M, tau(v) >= d(v) - d(M)(v) + 1, where d(M)(v) is the degree of v in M. Resistant subgraphs have key role in the study of (irreversible) dynamic monopolies in graphs. Denote the smallest cardinality of any tau-resistant subset of G by rho(tau)(G). For any positive integer s with vertical bar G vertical bar <= s <= 2 vertical bar E(G)vertical bar, we define rho*s(G) as rho*(s)(G) = max(tau:(Sigma u is an element of v(G)tau(u)) = s) rho(tau)(G). In this paper we first investigate the relationship between rho*(s) (G) and the well-known concept defensive alliance and then we study some properties of rho*(s)(G) as a function in s. At the end we determine p*(s)(K-n) for any value s and also specify the minimum resistant sets in K-n.