Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Saturated Partial Embeddings of Planar Graphs

Alexander Clifton, Nika Salia

Abstract

In this work, we study how far one can deviate from optimal behavior when embedding a planar graph. For a planar graph $G$, we say that a plane subgraph $H\subseteq G$ is a \textit{plane-saturated subgraph} if adding any edge (possibly with new vertices) to $H$ would either violate planarity or make the resulting graph no longer a subgraph of $G$. For a planar graph $G$, we define the \textit{plane-saturation ratio}, $\psr(G)$, as the minimum value of $\frac{e(H)}{e(G)}$ for a plane-saturated subgraph $H$ of $G$ and investigate how small $\psr(G)$ can be. While there exist planar graphs where $\psr(G)$ is arbitrarily close to $0$, we show that for all twin-free planar graphs, $\psr(G)>1/16$, and that there exist twin-free planar graphs where $\psr(G)$ is arbitrarily close to $1/16$. In fact, we study a broader category of planar graphs, focusing on classes characterized by a bounded number of degree $1$ and degree $2$ twin vertices. We offer solutions for some instances of bounds while positing conjectures for the remaining ones.

Saturated Partial Embeddings of Planar Graphs

Abstract

In this work, we study how far one can deviate from optimal behavior when embedding a planar graph. For a planar graph , we say that a plane subgraph is a \textit{plane-saturated subgraph} if adding any edge (possibly with new vertices) to would either violate planarity or make the resulting graph no longer a subgraph of . For a planar graph , we define the \textit{plane-saturation ratio}, , as the minimum value of for a plane-saturated subgraph of and investigate how small can be. While there exist planar graphs where is arbitrarily close to , we show that for all twin-free planar graphs, , and that there exist twin-free planar graphs where is arbitrarily close to . In fact, we study a broader category of planar graphs, focusing on classes characterized by a bounded number of degree and degree twin vertices. We offer solutions for some instances of bounds while positing conjectures for the remaining ones.
Paper Structure (12 sections, 10 theorems, 48 equations, 1 figure)

This paper contains 12 sections, 10 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:Twin-free']}
  3. Construction of a twin-free graph achieving the desired upper bound
  4. Lower Bound Proof
  5. Preliminaries
  6. Establishing bounds on the number of edges of Lg and Lg
  7. Proof when Lg
  8. Proof when Lg
  9. Proof of Theorem \ref{['thm:mostK']}
  10. Modifying the upper bound construction
  11. Modifying the lower bound proof
  12. Conclusion

Key Result

Theorem 1.3

For every planar graph $G$ with no degree $1$ or degree $2$ twins, For every positive $\epsilon$, there exists a twin-free planar $G_{\varepsilon}$ such that

Figures (1)

  • Figure 1: On the left, graph $G_{2n+5}$ and on the right, graph $H_{2n+5}$ from Example \ref{['ex:onefifth']}.

Theorems & Definitions (27)

  • Example 1.1
  • Definition
  • Example 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Claim 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 17 more