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Improved Outerplanarity Bounds for Planar Graphs

Therese Biedl, Debajyoti Mondal

TL;DR

This work studies the outerplanarity of planar graphs and introduces fence-girth $g$ as a central parameter that captures the length of the shortest fence separating inner and outer regions. Using a tree-of-peels decomposition, an augmentation $H$ preserving peels, and a detour-method to connect vertices, the authors derive upper bounds that scale as $\left\lfloor \frac{n-2}{2g} \right\rfloor + O(g)$ and relate outerplanarity to the graph diameter via $\text{outerplanarity} \le \tfrac{1}{2}\,\text{diam}(G) + O(\sqrt{n})$. For triangulated graphs they obtain a radius bound $\text{rad}(G) \le \frac{n}{2\kappa} + O(1)$ and provide a linear-time witness algorithm; in bipartite planar graphs the bound improves to $\frac{n}{8} + O(1)$ due to larger fence-girth. All bounds are tight up to lower-order terms, and the authors present linear-time constructions to achieve the bounds. These results have practical impact on planar graph drawing and on deriving fast algorithms for planar problems by controlling outerplanarity via fence-girth and diameter.

Abstract

In this paper, we study the outerplanarity of planar graphs, i.e., the number of times that we must (in a planar embedding that we can initially freely choose) remove the outerface vertices until the graph is empty. It is well-known that there are $n$-vertex graphs with outerplanarity $\tfrac{n}{6}+Θ(1)$, and not difficult to show that the outerplanarity can never be bigger. We give here improved bounds of the form $\tfrac{n}{2g}+2g+O(1)$, where $g$ is the fence-girth, i.e., the length of the shortest cycle with vertices on both sides. This parameter $g$ is at least the connectivity of the graph, and often bigger; for example, our results imply that planar bipartite graphs have outerplanarity $\tfrac{n}{8}+O(1)$. We also show that the outerplanarity of a planar graph $G$ is at most $\tfrac{1}{2}$diam$(G)+O(\sqrt{n})$, where diam$(G)$ is the diameter of the graph. All our bounds are tight up to smaller-order terms, and a planar embedding that achieves the outerplanarity bound can be found in linear time.

Improved Outerplanarity Bounds for Planar Graphs

TL;DR

This work studies the outerplanarity of planar graphs and introduces fence-girth as a central parameter that captures the length of the shortest fence separating inner and outer regions. Using a tree-of-peels decomposition, an augmentation preserving peels, and a detour-method to connect vertices, the authors derive upper bounds that scale as and relate outerplanarity to the graph diameter via . For triangulated graphs they obtain a radius bound and provide a linear-time witness algorithm; in bipartite planar graphs the bound improves to due to larger fence-girth. All bounds are tight up to lower-order terms, and the authors present linear-time constructions to achieve the bounds. These results have practical impact on planar graph drawing and on deriving fast algorithms for planar problems by controlling outerplanarity via fence-girth and diameter.

Abstract

In this paper, we study the outerplanarity of planar graphs, i.e., the number of times that we must (in a planar embedding that we can initially freely choose) remove the outerface vertices until the graph is empty. It is well-known that there are -vertex graphs with outerplanarity , and not difficult to show that the outerplanarity can never be bigger. We give here improved bounds of the form , where is the fence-girth, i.e., the length of the shortest cycle with vertices on both sides. This parameter is at least the connectivity of the graph, and often bigger; for example, our results imply that planar bipartite graphs have outerplanarity . We also show that the outerplanarity of a planar graph is at most diam, where diam is the diameter of the graph. All our bounds are tight up to smaller-order terms, and a planar embedding that achieves the outerplanarity bound can be found in linear time.
Paper Structure (15 sections, 9 theorems, 5 equations, 6 figures)

This paper contains 15 sections, 9 theorems, 5 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Toolbox
  4. The tree of peels:
  5. Augmenting $G$:
  6. The detour-method:
  7. Outerplanarity and $\text{fence-girth}$
  8. Separator-node $S$:
  9. Vertices stored in ${\cal T}_R$:
  10. Vertices at descendants of $S$:
  11. Discussion:
  12. Run-time considerations
  13. Tightness
  14. Outerplanarity and Diameter
  15. Remarks

Key Result

theorem thmcountertheorem

LT79 Let ${\cal T}$ be a tree with non-negative node-weights $w(\cdot)$. Then in linear time we can find a node $S$ such that for every subtree ${\cal T}'$ of ${\cal T}\setminus S$ we have $w({\cal T}')\leq \tfrac{1}{2}w({\cal T})$, where $w({\cal T}')$ denotes the sum of weights of nodes in ${\cal

Figures (6)

  • Figure 1: (a) A plane graph $G$. (b)--(e) The graphs obtained by deleting $L_0,L_1,\ldots, L_3$, respectively. Solid vertices are the set $V(N)$ of the corresponding node $N$. (f) The tree of peels $\cal T$. (g) The augmentation $H$.
  • Figure 2: Larger example of $\cal T$ and the detour-method to connect $s_0\in A_0$ with $z_0\in A_0$, as well as $s$ to $z$ as in Claim \ref{['cl:caseRoot']}. Not all directed edges are shown for clarity.
  • Figure 3: (a) Path-finding for nodes in $V({\cal T}_R)$when $Z$ is not a descendent of $S$. (b) Path-finding by using a detour at $S$when $Z$ is a descendent of $S$. (c) Path-finding for deep descendants of $S$. (d) Finding a small common ancestor of deep nodes.
  • Figure 4: Graphs that have large outerplanarity relative to the $\text{fence-girth}$.
  • Figure 5: Graphs $G^4_3$, $H^4_3$ and $H^8_3$.
  • ...and 1 more figures

Theorems & Definitions (30)

  • proof
  • theorem thmcountertheorem
  • proof
  • proof
  • proof
  • definition thmcounterdefinition
  • proof
  • lemma thmcounterlemma
  • proof
  • proof
  • ...and 20 more