On the Path-Width of Planar Graphs (original) (raw)
Related papers
Approximation of pathwidth of outerplanar graphs
Journal of Algorithms, 2002
There exists a polynomial time algorithm to compute the pathwidth of outerplanar graphs [3], 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 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.
On the structure of graphs with path-width at most two
2012
Nancy G. Kinnersley and Michael A. Langston has determined the excluded minors for the class of graphs with pathwidth at most two by computer. Their list consisted of 110 graphs. Such a long list is difficult to handle and gives no insight to structural properties. We take a different route, and concentrate on the building blocks and how they are glued together. In this way, we get a characterization of 2-connected and 2-edge-connected graphs with path-width at most two. Along similar lines, we sketch the complete characterization of graphs with path-width at most two.
Paths of low weight in planar graphs
Discussiones Mathematicae Graph Theory, 2008
The existence of paths of low degree sum of their vertices in planar graphs is investigated. The main results of the paper are:
A note on planar graphs with large width parameters and small grid-minors
Discrete Applied Mathematics, 2012
Given a graph G with tree-width ω(G), branch-width β(G), and side size of the largest square grid-minor θ (G), it is known that θ (G) ≤ β(G) ≤ ω(G) + 1 ≤ 3 2 β(G). In this paper, we introduce another approach to bound the side size of the largest square gridminor specifically for planar graphs. The approach is based on measuring the distances between the faces in an embedding of a planar graph. We analyze the tightness of all derived bounds. In particular, we present a class of planar graphs where θ (G) = β(G) < ω(G) = ⌊ 3 2 θ (G)⌋ − 1.
The radius of -connected planar graphs with bounded faces
2012
We prove that if G is a 3-connected plane graph of order p, maximum face length l and radius rad(G), then the bound rad(G) ≤ p 6 + 5l 6 + 2 3 holds. For constant l, our bound is shown to be asymptotically sharp and improves on a bound by Harant (1990) [6]. Furthermore we extend these results to 4-and 5-connected planar graphs.
A bound on the treewidth of planar even-hole-free graphs
Discrete Applied Mathematics, 2010
The class of planar graphs has unbounded treewidth, since the k × k grid, ∀k ∈ N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an even-hole-free planar graph, then it does not contain a 9×9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 49.
Even-hole-free planar graphs have bounded treewidth
Electronic Notes in Discrete Mathematics, 2008
The class of planar graphs has unbounded treewidth, since the k × k grid, ∀k ∈ N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an evenhole-free planar graph, then it does not contain a 9 × 9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 44.
The Electronic Journal of Combinatorics, 2018
A nonplanar graph GGG is called almost-planar if for every edge eee of GGG, at least one of GbackslasheG\backslash eGbackslashe and G/eG/eG/e is planar. In 1990, Gubser characterized 3-connected almost-planar graphs in his dissertation. However, his proof is so long that only a small portion of it was published. The main purpose of this paper is to provide a short proof of this result. We also discuss the structure of almost-planar graphs that are not 3-connected.
Path Choosability of Planar Graphs
The Electronic Journal of Combinatorics, 2018
A path coloring of a graph GGG is a vertex coloring of GGG such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of 333 colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size 222.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of qqq colors. We wish to color each vertex with rrr colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if q/rge2q/r \ge 2q/rge2. For planar graphs we conjecture that a similar result holds with q/rge3q/r \ge 3q/rge3; we present partial results toward this conjecture.
On a graph property generalizing planarity and flatness
Combinatorica, 2009
We introduce a topological graph parameter σ(G), defined for any graph G. This parameter characterizes subgraphs of paths, outerplanar graphs, planar graphs, and graphs that have a flat embedding as those graphs G with σ(G) ≤ 1, 2, 3, and 4, respectively. Among several other theorems, we show that if H is a minor of G, then σ(H) ≤ σ(G), that σ(Kn) = n−1, and that if H is the suspension of G, then σ(H) = σ(G)+1. Furthermore, we show that µ(G) ≤ σ(G) + 2 for each graph G. Here µ(G) is the graph parameter introduced by Colin de Verdière in [2].