Network vulnerability parameter and results on two surfaces

Isolation toughness is a vital parameter to evaluate the vulnerability of computer networks. In specific network designing stage, it is necessary to find the lower bound of the isolated toughness, and strive to build a network that meets the stability requirements with the least cost. Gao et al.1 conjectured that if a graph G with κ ( G ) ≥ 3 m + 1 2 satisfies I ( G ) > 7 m + 5 4 m + 4 or I ′ ( G ) > 7 m + 5 4 m + 2 , then G is a ( P ≥ 3 , m ) -factor deleted graph. It's proved that this conjecture holds. However, it is found that as the connectivity changes, the tight lower bound of isolated toughness for ( P ≥ 3 , m ) -factor deleted graphs will change as well. Therefore, we propose a new perspective to look into this problem and introduce the concepts of isolated toughness ( P ≥ 3 , m ) factor deleted surface and isolated toughness variant ( P ≥ 3 , m ) factor deleted surface, where the result of the original conjecture is only a cross-section on surfaces. The main contribution in this paper is to determine the concrete expression of these two surfaces.