Grunnleggende konsepter
Every degree sequence in the family D with even sum between 2n and 4n-6-2ω1 can be realized by either a provably non-outerplanar graph or a graph with a 2-page book embedding, one of whose pages is also bipartite.
Sammendrag
The paper studies the problem of determining whether a given sequence of positive integers d = (d1, d2, ..., dn) is the degree sequence of some outerplanar (1-page book embeddable) graph G.
The key results are:
If the sum of the degrees ∑d is at most 2n-2, then d has a realization by a forest, which is trivially outerplanar.
The authors focus on the family D of sequences d with even sum 2n ≤ ∑d ≤ 4n-6-2ω1, where ωx is the number of x's in d. This is because ∑d > 2n-2 is a necessary condition for a sequence to be non-forestic (i.e., not realizable by a forest).
The authors partition D into two disjoint subfamilies, DNOP and D2PBE:
DNOP contains sequences that are provably non-outerplanar.
D2PBE contains sequences that are given a realizing graph G enjoying a 2-page book embedding, where one of the pages is also bipartite.
For sequences d in D≤4 where the maximum degree d1 ≤ 4, the authors show that d has an OP+2 realization, where one page is outerplanar and the other page consists of at most two edges.
For sequences d in D≥5 where d1 ≥ 5, the authors provide a more complex analysis. They show that if d satisfies certain conditions, including ∑d = 4n-6, 2ω2 + ω3 ≤ n+1, and ω2 > 2, then d has an OP+bi realization, where one page is outerplanar and the other page is bipartite.