Approximate Realizations for Outerplanaric Degree Sequences

TL;DR

This paper investigates the conditions under which degree sequences can be realized by outerplanar graphs, providing classifications and explicit constructions for sequences within specific sum ranges.

Contribution

It introduces a partition of degree sequences into non-outerplanar and realizable classes with 2-page embeddings, advancing understanding of outerplanar degree sequence realizability.

Findings

Sequences with sum ≤ 2n - 2 are realizable as forests.

Sequences in a specific range are either non-outerplanar or have 2-page bipartite embeddings.

The paper provides a classification and constructive methods for these sequences.

Abstract

We study the question of whether a sequence d = (d_1,d_2, \ldots, d_n) 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 \sum d \leq 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 \cD of all sequences d of even sum 2n\leq \sum d \le 4n-6-2\multipl_1, where \multipl_x is the number of x's in d. (The second inequality is a necessary condition for a sequence d with \sum d\geq 2n to be outerplanaric.) We partition \cD into two disjoint subfamilies, \cD=\cD_{NOP}\cup\cD_{2PBE}, such that every sequence in \cD_{NOP} is provably non-outerplanaric, and every sequence in \cD_{2PBE} is given a realizing graph $G$ enjoying a 2-page book embedding (and moreover, one of the pages is also bipartite).