Approximate Realizations for Outerplanaric Degree Sequences

TL;DR

This work tackles the problem of deciding when a degree sequence is outerplanar realizable and constructing such realizations. It introduces the candidate family \mathcal{D} and partitions it into \mathcal{D}_{NOP} (provably non-outerplanar) and \mathcal{D}_{2PBE} (admitting a 2-page embedding with a bipartite second page), providing a polynomial-time classification and constructive procedures. The main contribution is a detailed, case-based framework that either certifies non-outerplanarity or produces a 2PBE realization (OP+bi) for sequences in \mathcal{D}, with stronger realizations (OP on one page) achievable in many instances. This yields an effective, approximate algorithmic approach to outerplanar degree realization and advances the understanding of degree-sequence realizability within outerplanar and related graph families.

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).

Figures (14)

• Figure 1: A flowchart of the resulting algorithm. $\Delta_{E}= (4n - 6 - \sum d) / 2$ and $\Delta_{\omega}=3\omega_1+2 \omega_2 + \omega_3 - n$.
• Figure 2: An outerplanar graph without degree-1 vertices. The triangle $(5,7,9)$ is the only internal cycle, while $(1,2,3)$, $(3,4,5,6,7,8,9,10)$, and $(11,12,13)$ are outer-cycles. A possible cyclic-order is $(1,2,3,4,5,6,7,8,9,10,11,12,13)$.
• Figure 3: A caterpillar graph with degree sequence $(5,4^3,2,1^{11})$. Spine vertices are gray, leaves are yellow.
• Figure 4: A merged sequence $R$ and its corresponding bracket assignment.
• Figure 5: Illustration of $G$ as constructed in Lemma \ref{['lem:OP-4n-1']}. The vertices $x_1, \ldots,x_6$ and the leaves in yellow (labels omitted) together with the black edges form the caterpillar graph $G'$. Note that $x_5$ has no leaves. Paths $P_1$ and $P_2$ are depicted in green and blue, respectively. Together, they form the cycle $C$.

Theorems & Definitions (46)

• proof
• Lemma 2: West01
• Lemma 3: syslo1979characterizations
• Lemma 4
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• Lemma 5
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• Lemma 6
• Lemma 7
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