Approximate realizations for outerplanaric degree sequences

Abstract

We study the question of whether a sequence $d=(d_1,\dots ,d_n)$ 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 $\sum d\le 2n-2$ 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 $2n\le \sum d\le 4n-6-2{\omega }_1$, where ${\omega }_x$ is the number of x's in d. We partition $D$ into two disjoint subfamilies, $D=D_{NOP}\cup D_{2PBE}$, 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).