Approximate Realizations for Outerplanaric Degree Sequences

We study the question of whether a sequence d = (d₁, d₂, , dₙ) of positive integers is the degree sequence of some outerplanar (a. k. a. 1-page book embeddable) 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 (which is trivially an outerplanar graph). In this paper, we consider the family of all sequences d of even sum 2n d 4n-6-2₁, where ₓ is the number of x's in d. (The second inequality is a necessary condition for a sequence d with d 2n to be outerplanaric. ) We partition into two disjoint subfamilies, =₍₎₏₂₏₁₄, such that every sequence in ₍₎₏ is provably non-outerplanaric, and every sequence in ₂₏₁₄ is given a realizing graph G enjoying a 2-page book embedding (and moreover, one of the pages is also bipartite).