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ISSN 0249-6399

ISRN INRIA/RR--????--FR+ENG

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

On the Pathwidth of Planar Graphs

Omid Amini — Florian Huc — Stéphane Pérennes

N° ????

Juin 2006

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Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 —Télécopie : +33 4 92 38 77 65

On the Pathwidth of Planar Graphs

Omid Amini∗ †, Florian Huc∗ ‡, Stéphane Pérennes∗

Thème COM — Systèmes communicants

Projets MASCOTTE

Rapport de recherche n° ???? — Juin 2006 — 6 pages

Abstract: Fomin and Thilikos in [5] conjectured that there is a constant c such that, for every 2-

connected planar graph G, pw(G∗) ≤ 2pw(G)+c (the same question was asked simutaneously by Coudert,

Huc and Sereni in [4]). By the results of Boedlander and Fomin [2] this holds for every outerplanar graph

and actually is tight by Coudert, Huc and Sereni [4]. In [5], Fomin and Thilikos proved that there is a

constant c such that the pathwidth of every 3-connected graph G satisfies: pw(G∗) ≤ 6pw(G) + c. In

this paper we improve this result by showing that the dual a 3-connected planar graph has pathwidth at

most 3 times the pathwidth of the primal plus two. We prove also that the question can be answered

positively for 4-connected planar graphs.

Key-words:

planar graphs, pathwidth

∗Projet Mascotte (cnrs, inria, unsa), INRIA Sophia-Antipolis, 2004 route des Lucioles BP 93, 06902 Sophia-Antipolis

Cedex, France

†École Polytechnique, 91120 Palaiseau

‡This author is partialy supported by Région Provence Alpes Côte d’Azur

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A Propos de la Pathwidth des Graphes Planaires

Résumé :

3-connexe est au plus 6 fois celle de son dual à une constante près, ont conjecturé que pour tout graphe

planaire biconnexe G, pw(G∗) ≤ 2pw(G) + cte. D’après Boedlander et Fomin [2] cela est vrai pour

tout graphe outerplanaire. De plus cela est exact d’après Coudert, Huc and Sereni [4]. Dans cet article

nous améliorons le résultat de Fomin et Thilikos en montrant que la pathwidth de tout graphe planaire

3-connexe est au plus 3 fois celle de son dual plus 2. Nous démontrons également que la conjecture est

vrai pour tout graphe planaire dont le dual est 4-connexe.

Fomin et Thilikos[5], après avoir démontré que la pathwidth de tout graphes planaires

Mots-clés :

graphes planaires, pathwidth

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On the Pathwidth of Planar Graphs

3

1 Introduction

A planar graph is a graph that can be embedded in the plane without crossing edges. It is said to be

outerplanar if it can be embedded in the plane without crossing edges and such that all its vertices are

incident to the unbounded face. For any graph G, we denote by V (G) its vertex set and by E(G) its

edge set. The dual of the planar graph G, denoted by G∗, is the graph obtained by putting one vertex

for each face, and joining two vertices if and only if the corresponding faces are adjacent. Note that the

dual of a planar graph can also be computed in linear time.

The notion of pathwidth was introduced by Robertson and Seymour [9]. A path decomposition of a

graph G = (V,E) is a set system (X1,...,Xr) of V such that

?r

2. ∀xy ∈ E,∃i ∈ {1,...,r} : {x,y} ⊂ Xi;

3. for all 1 ≤ i0< i1< i2≤ r, Xi0∩ Xi2⊆ Xi1.

The width of the path decomposition (X1,...,Xr) is max1≤i≤r|Xi|−1. The pathwidth of G, denoted by

pw(G), is the minimum width over its path decompositions.

Computing the pathwidth of graphs is an active research area, in which a lot of work has been

done (Fer a survay see for instance [8]). Govindan et al. [6] gave an O(nlog(n)) time algorithm for

approximating the pathwidth of outerplanar graphs with a multiplicative factor of 3. For biconnected

outerplanar graphs, Bodlaender and Fomin [2] improved upon this result by giving a linear time algorithm

which approximates the pathwidth of biconnected outerplanar graphs with a multiplicative factor 2. To

do so, they exhibit a relationship between the pathwidth of an outerplanar graph and the pathwidth of

its dual. More precisely, the following holds.

1.

i=1Xi= V ;

Theorem 1 (Bodlaender and Fomin [2]) Let G be a biconnected outerplanar graph without loops and

multiple edges. Then pw(G∗) ≤ pw(G) ≤ 2pw(G∗) + 2.

Since the weak dual of an outerplanar graph (which can be computed in linear time) is a tree and there

exist linear time algorithms to compute the pathwidth of a tree [11], this yields the desired approximation.

Coudert, Huc and Sereni in [4] improved this result by proving the following theorem:

Theorem 2 (Coudert, Huc and Sereni [4]) For every biconnected outerplanar graph G, we have

pw(G∗) ≤ pw(G) ≤ 2 pw(G∗) − 1 and all the values in the interval [pw(G∗) , 2 pw(G∗) − 1] can be

the pathwidth of G.

Simultaneously Coudert, Huc and Sereni state the following question as an open problem in [4] and

Fomin and Thilikos conjectured it in [5] :

Conjecture 1 ([5],[4]) Is there a constant c such that, for every 2-connected planar graph G,1

c ≤ pw(G) ≤ 2pw(G∗) + c?

It is worth noting that this conjecture is motivated by the following result about the treewidth,

conjectured by Robertson and Seymour [10] and proved by Lapoire [7] using algebraic methods (notice

that Bouchitté, Mazoit and Todinca [3] gave a shorter and combinatorial proof of this result).

2pw(G∗)−

Theorem 3 ([7]) For every planar graph G, tw(G) ≤ tw(G∗) + 1.

Fomin and Thilikos made an even stronger conjecture :

Conjecture 2 ([5]) There is a constant c such that for every 2-connected planar graphs G of treewidth

at least m, pw(G∗) ≤

m

m−1pw(G) + c

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