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Ranking and Unranking of the Planar Embeddings of a Planar Graph

Giuseppe Di Battista, Fabrizio Grosso, Giulia Maragno, Maurizio Patrignani

Abstract

Let $\mathcal{G}$ be the set of all the planar embeddings of a (not necessarily connected) $n$-vertex graph $G$. We present a bijection $Φ$ from $\mathcal{G}$ to the natural numbers in the interval $[0 \dots |\mathcal{G}| - 1]$. Given a planar embedding $\mathcal{E}$ of $G$, we show that $Φ(\mathcal{E})$ can be decomposed into a sequence of $O(n)$ natural numbers each describing a specific feature of $\mathcal{E}$. The function $Φ$, which is a ranking function for $\mathcal{G}$, can be computed in $O(n)$ time, while its inverse unranking function $Φ^{-1}$ can be computed in $O(n α(n))$ time. The results of this paper can be of practical use to uniformly at random generating the planar embeddings of a graph $G$ or to enumerating such embeddings with amortized constant delay. Also, they can be used to counting, enumerating or uniformly at random generating constrained planar embeddings of $G$.

Ranking and Unranking of the Planar Embeddings of a Planar Graph

Abstract

Let be the set of all the planar embeddings of a (not necessarily connected) -vertex graph . We present a bijection from to the natural numbers in the interval . Given a planar embedding of , we show that can be decomposed into a sequence of natural numbers each describing a specific feature of . The function , which is a ranking function for , can be computed in time, while its inverse unranking function can be computed in time. The results of this paper can be of practical use to uniformly at random generating the planar embeddings of a graph or to enumerating such embeddings with amortized constant delay. Also, they can be used to counting, enumerating or uniformly at random generating constrained planar embeddings of .

Paper Structure

This paper contains 17 sections, 12 theorems, 1 equation, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simply Connected Graphs
  4. Transforming a tuple into an embedding
  5. Transforming an embedding into a tuple
  6. Non-Connected Graphs
  7. Conclusions and Future Work
  8. A full example of ranking and unranking
  9. Graph Decomposition
  10. Block-cutvertex Tree
  11. SPQR-tree
  12. Prüfer sequence
  13. How to rank a tuple
  14. Biconnected Graphs
  15. Connected Graphs
  16. ...and 2 more sections

Key Result

theorem 1

Let $G$ be an $n$-vertex planar graph with connected components $G_1, \dots, G_t$. Let $v_1, \dots, v_w$ be the cut-vertices of $G$. Let $P_1, \dots, P_y$ be the parallel triconnected components of the SPQR-trees of all the biconnected components of $G$. Let $R_1, \dots, R_z$ be the rigid triconnect where The number of elements of the tuples are $O(n)$. The function $\Phi$ can be computed in $O(n

Figures (12)

  • Figure 1: The unranking of the embedding 6-1-8.
  • Figure 2: The planar embedding on a sphere around the cut-vertex $v$ and its resulting nesting-tree. The ranking will compute the numbers 6-1-8.
  • Figure 3: A graph drawn according to its $754.705.812.645^{th}$ planar embedding on the sphere, corresponding to the sequence $\langle 0,11,1,0,1,0,0,0,1,0,1,1,0,1, 2,1,6,1,4,5,0,1 \rangle$, as defined in \ref{['th:general']}. In particular, the bound for each number, as described in \ref{['le:tuple-number']}, is $\langle 17,17,17,2,9,8,1,2,3,1,2,2,2,4,9,8,7,2,6,6,2,2 \rangle$ and the number of embeddings is 19.716.667.342.848. Also, the rank restricted to the red (resp. green) biconnected subgraph is 21 (resp. 4), and the rank of the embedding around the cut-vertex 9 (in blue) is 3653.
  • Figure 4: The ranking of a rooted labeled tree using the Prüfer sequence.
  • Figure 5: The unranking of a rooted labeled tree using the Prüfer sequence $4,1,5,4,5$.
  • ...and 7 more figures

Theorems & Definitions (17)

  • theorem 1
  • lemma 1
  • theorem 2
  • theorem 3
  • lemma 2
  • lemma 3
  • lemma 4
  • theorem 4
  • proof
  • lemma 4
  • ...and 7 more