Approximate realizations for outerplanaric degree sequences

We study the question of whether a sequence d=(d₁,…,d_n) of positive integers is the degree sequence of some outerplanar graph G. If so, G is an outerplanar realization of d and d is an outerplanaric sequence. The case where ∑d≤2n−2 is easy, as d has a realization by a forest. In this paper, we consider the family D of all sequences d of even sum 2n≤∑d≤4n−6−2ω₁, where ω_x is the number of x's in d. We partition D into two disjoint subfamilies, D=D_NOP∪D_2PBE, such that every sequence in D_NOP is provably non-outerplanaric, and every sequence in D_2PBE is given a realizing graph G enjoying a 2-page book embedding (and moreover, one of the pages is also bipartite).