Gröbner bases method for solving N-path in finite graph and its application

Let G be a finite directed graph with no loop and no heavy edges, or an undirected graph with no loops and no edges. 𝑁 is a given natural number. This paper proves that the existence problem of two paths with length 𝑁 in G (referred to as 𝑁-path) is completely equivalent to the solutions of a multivariate of polynomial 𝑁 the range of {0,1} or {0,1, −1}. Therefore, the Gröbner bases method can be used to give an effective discrimination of the existence of the solution. This result can be applied to solve the problems of cutting edge judgment, tree and forest judgment in G.