Let G^D be the set of graphs G(V, E) with n vertices, and the degree sequence equal to D = (d₁, d₂,..., d_n). In addition, for ½ < a < 1, we define the set of graphs with an almost given degree sequence D as follows:

G_a(D): = ∪ G(D'),

where the union is over all degree sequences D¯ such that, for 1 ≤ i ≤ n, we have |d _i-d'_i|< (d_i)^a.

Now, we choose random graphs G_g(D) and G_a(D) uniformly out of the sets G(D) and G_a(D) , respectively, which we call random graphs with given and almost given degree sequence D. We first survey the known results about graphs and random graphs with a given degree sequence. This has been studied when G_g(D) is a dense graph, i.e. |E|=Θ(n²), in the sense of graphons, or when G_g(D) is very sparse, i.e. (d_n)²=o(|E|).

Second and in the case of sparse graphs with an almost given degree sequence, we investigate this question, and give the finite tree subgraph structure of G_a(D) under some mild conditions. For the random graph G_g(D) with a given degree sequence, we re-derive the finite tree structure in dense and very sparse cases to give a continuous picture.

Finally, for a pair of integer vectors (D₁, D ₂) ∈ Z^(n₁) × Z ^(n₂), we let G_b(D₁, D₂) be the random bipartite graph that is chosen uniformly out of the set G(D₁, D₂), where G(D₁, D₂) is the set of all bipartite graphs with the degree sequence (D₁, D₂). We are able to show the result for G_b(D₁, D ₂) without any further conditions.