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## Analysis of the impact of the number of edges in connected graphs on the computational complexity of the independent set problem

Journal of Applied and Industrial Mathematics. 2012. Vol. 6. No. 1. P. 97-99.

Under study is the complexity status of the independent set problem in a class of connected graphs that are defined by functional constraints on the number of edges depending on the number of vertices. For every natural number C, this problem is shown to be polynomially solvable in the class of graphs, On the other hand, this problem is proved to be not polynomially solvable in the class of graphs, for every unbounded nondecreasing function f(n): ℕ → ℕ, such that exponent of this function grows faster than a polynomial function of n. The latter result is true if there does not exist a subexponential algorithm for solving the independent set problem.