Constructing Outerplanar and 2-Page Book Embeddings for Degree Sequences

Kernkonzepte

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.

Zusammenfassung

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.

Statistiken

None

Zitate

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Wichtige Erkenntnisse aus

Approximate Realizations for Outerplanaric Degree Sequences

by Amotz Bar-No... um arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03278.pdf

Tiefere Fragen

Can the authors' approach be extended to characterize the outerplanarity of all degree sequences in the family D, not just an approximate solution

The authors' approach can potentially be extended to characterize the outerplanarity of all degree sequences in the family D, not just an approximate solution. By refining the classification criteria and exploring more intricate relationships between the degrees of vertices in outerplanar graphs, it may be possible to develop a more comprehensive algorithm that can determine the outerplanarity of any given degree sequence in D with certainty. This could involve further analyzing the structural properties of outerplanar graphs and identifying specific patterns or constraints that uniquely define outerplanaric sequences.

What are the implications of the 2-page book embedding results for practical applications, such as graph visualization or network design

The results regarding 2-page book embeddings have significant implications for practical applications in graph visualization and network design. By providing a method to represent outerplanar graphs in a 2-page book embedding format, the research offers a visually intuitive way to understand and analyze the connectivity and structure of such graphs. In graph visualization, this can lead to more efficient and effective ways to display and interpret complex network data. Additionally, in network design, the ability to embed graphs in a 2-page book format can aid in optimizing the layout and organization of network components, potentially improving the overall performance and efficiency of network systems.

Are there any connections between the properties of outerplanar graphs and other well-studied graph classes, such as planar graphs or 2-trees, that could provide further insights

There are several connections between the properties of outerplanar graphs and other well-studied graph classes that could provide further insights. For example, outerplanar graphs are a subset of planar graphs, as every outerplanar graph can be embedded in the plane without edge crossings. This relationship highlights the similarities and differences between these two graph classes, offering opportunities to explore the boundary cases and transitions between them. Additionally, the concept of 2-trees, which are graphs that can be built from a single vertex by adding edges in a tree-like structure, can be related to outerplanar graphs through their shared properties of acyclicity and planarity. By studying these connections, researchers can gain a deeper understanding of the structural characteristics and behaviors of different graph classes.