Approximation of pathwidth of outerplanar graphs

doi:10.1016/S0196-6774(02)00001-9

Journal of Algorithms

Volume 43, Issue 2, May 2002, Pages 190-200

https://doi.org/10.1016/S0196-6774(02)00001-9 Get rights and content

Abstract

There exists a polynomial time algorithm to compute the pathwidth of outerplanar graphs, but the large exponent makes this algorithm impractical. In this paper, we give an algorithm that, given a biconnected outerplanar graph G, finds a path decomposition of G of pathwidth at most twice the pathwidth of G plus one. To obtain the result, several relations between the pathwidth of a biconnected outerplanar graph and its dual are established.

References (10)

N. Robertson et al.
Graph minors. I. Excluding a forest
J. Combin. Theory Ser. B
(1983)
H.L. Bodlaender
A partial k-arboretum of graphs with bounded treewidth
Theor. Comp. Sci.
(1998)
H.L. Bodlaender et al.
Efficient and constructive algorithms for the pathwidth and treewidth of graphs
J. Algorithms
(1996)
J.A. Ellis et al.
The vertex separation and search number of a graph
Inform. and Comput.
(1994)
S.L. Mitchell
Linear algorithms to recognize outerplanar and maximal outerplanar graphs
Inform. Process. Lett.
(1979)

There are more references available in the full text version of this article.

Cited by (29)

Branchwidth is (1,g)-self-dual
2024, Discrete Applied Mathematics
A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. We prove that, in the class of connected hypergraphs without bridges and loops that are embeddable in some surface of Euler genus at most $g$ , branchwidth is a $(1, g)$ -self-dual parameter, i.e., for every hypergraph $G$ in the class, the branchwidth of its dual is at most the branchwidth of $G$ plus $g$ . This is the first proof that branchwidth is an additively self-dual width parameter.
Tree t-spanners in outerplanar graphs via supply demand partition
2015, Discrete Applied Mathematics
A tree $t$ -spanner of an unweighted graph $G$ is a spanning tree $T$ such that for every two vertices their distance in $T$ is at most $t$ times their distance in $G$ . Given an unweighted graph $G$ and a positive integer $t$ as input, the Tree $t$ -spanner problem is to compute a tree $t$ -spanner of $G$ if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar graphs. We present a linear-time reduction from Tree $t$ -spanner in outerplanar graphs to the supply–demand tree partition problem. Based on this reduction, we obtain a linear-time algorithm to solve Tree $t$ -spanner in outerplanar graphs. Consequently, we show that the minimum value of $t$ for which an input outerplanar graph on $n$ vertices has a tree $t$ -spanner can be found in $O (n log n)$ time.
Non-deterministic graph searching in trees
2015, Theoretical Computer Science
Citation Excerpt :
Ellis and Markov [17] gave a linear time algorithm for the class of unicyclic graphs. Bodlaender and Fomin [18] and Coudert et al. [19] use the weak dual of an outerplanar graph, that is a tree, to approximate its pathwidth. A restricted (non-deterministic search) strategy is a monotone non-deterministic search strategy such that the moves are ordered in the following particular way.
Non-deterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given $q \geq 0$ , the q-limited search number, $s_{q} (G)$ , of a graph G is the smallest number of searchers required to capture an invisible fugitive in G, when the searchers are allowed to know the position of the fugitive at most q times. The search parameter $s_{0} (G)$ corresponds to the pathwidth of a graph G, and $s_{\infty} (G)$ to its treewidth. Determining $s_{q} (G)$ is NP-complete for any fixed $q \geq 0$ in general graphs and $s_{0} (T)$ can be computed in linear time in trees, however the complexity of the problem on trees has been unknown for any $q > 0$ .
We introduce a new variant of graph searching called restricted non-deterministic. The corresponding parameter is denoted by ${rs}_{q}$ and is shown to be equal to the non-deterministic graph searching parameter $s_{q}$ for $q = 0, 1$ , and at most twice $s_{q}$ for any $q \geq 2$ (for any graph G).
Our main result is a polynomial time algorithm that computes ${rs}_{q} (T)$ for any tree T and any $q \geq 0$ . This provides a 2-approximation of $s_{q} (T)$ for any tree T, and shows that the decision problem associated to $s_{1}$ is polynomial in the class of trees. Our proofs are based on a new decomposition technique for trees which might be of independent interest.
Approximate search strategies for weighted trees
2012, Theoretical Computer Science
The problems of (classical) searching and connected searching of weighted trees are known to be computationally hard. In this work we give a polynomial-time 3-approximation algorithm that finds a connected search strategy of a given weighted tree. This in particular yields constant factor approximation algorithms for the (non-connected) classical searching problems and for the weighted pathwidth problem for this class of graphs.
On self-duality of branchwidth in graphs of bounded genus
2011, Discrete Applied Mathematics
A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. Self-duality has been examined for several width parameters, such as branchwidth, pathwidth, and treewidth. In this paper, we give a direct proof of the self-duality of branchwidth in graphs embedded in some surface. In this direction, we prove that $bw (G^{*}) \leq 6 \cdot bw (G) + 2 g - 4$ for any graph $G$ embedded in a surface of Euler genus $g$ .
Linear-time algorithms for problems on planar graphs with fixed disk dimension
2007, Information Processing Letters
The disk dimension of a planar graph G is the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with a given disk dimension are derived, leading to an algorithm to obtain an outerplanar subgraph of a graph with disk dimension k by removing at most $2 k - 2$ vertices. This reduction is used to obtain linear-time exact and approximation algorithms on graphs with fixed disk dimension. In particular, a linear-time approximation algorithm is presented for the pathwidth problem.

View all citing articles on Scopus

^☆: This research was partially supported by EC contract IST-1999-14186: Project ALCOM-FT (Algorithms and Complexity–Future Technologies).

View full text

Approximation of pathwidth of outerplanar graphs☆

Abstract

J. Combin. Theory Ser. B

Theor. Comp. Sci.

J. Algorithms

Inform. and Comput.

Inform. Process. Lett.