Even-hole-free planar graphs have bounded treewidth

Discrete Mathematics, 1998

The oriented chromatic number o(H) of an oriented graph H is defined to be the minimum order of an oriented graph H' such that H has a homomorphism to H'. If each graph in a class ~ has a homomorphism to the same H', then H' is ~-universal. Let ~k denote the class of orientations of planar graphs with girth at least k. Clearly, ~3 ~ ~4 ~ ~5... We discuss the existence of ~k-universal graphs with special properties. It is known (see ) that there exists a ~3-universal graph on 80 vertices. We prove here that (1) there exist no planar ~4-universal graphs;

Siam Journal on Discrete Mathematics, 2009

Fomin and Thilikos in conjectured that there is a constant c such that, for every 2connected 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 , 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.

We show that every planar, 4-connected, K 2,5 -minorfree graph is the square of a cycle of even length at least six.

Journal of Graph Theory, 2016

The authors previously published an iterative process to generate a class of projectiveplanar K 3,4-free graphs called 'patch graphs'. They also showed that any simple, almost 4-connected, nonplanar, and projective-planar graph that is K 3,4-free is a subgraph of a patch graph. In this paper, we describe a simpler and more natural class of cubic K 3,4free projective-planar graphs which we call Möbius hyperladders. Furthermore, every simple, almost 4-connected, nonplanar, and projective-planar graph that is K 3,4-free is a minor of a Möbius hyperladder. As applications of these structures we determine the page number of patch graphs and of Möbius hyperladders.

Ars Mathematica Contemporanea

The acyclic number a(G) of a graph G is the maximum order of an induced forest in G. The purpose of this short paper is to propose a conjecture that a(G) ≥ 1 − 3 2g n holds for every planar graph G of girth g and order n, which captures three known conjectures on the topic. In support of this conjecture, we prove a weaker result that a(G) ≥ 1 − 3 g n holds. In addition, we give a construction showing that the constant 3 2 from the conjecture cannot be decreased.

Discrete Mathematics & Theoretical Computer Science, 2010

We proved that every planar triangle-free graph of order n has a subset of vertices that induces a forest of size at least (71n + 72)/128. This improves the earlier work of Salavatipour (2006). We also pose some questions regarding planar graphs of higher girth.

European Journal of Combinatorics, 2008

The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well-known that a(G) ≤ 3 for any planar graph G. In this paper we prove that a(G) ≤ 2 whenever G is planar and either G has no 4-cycles or any two triangles of G are at distance at least 3.

Journal of Graph Algorithms and Applications, 2009

A straight-line grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straight-line segment without edge crossings. It is well known that a planar graph of n vertices admits a straight-line grid drawing on a grid of area O(n 2). A lower bound of Ω(n 2) on the area-requirement for straight-line grid drawings of certain planar graphs are also known. In this paper, we introduce a fairly large class of planar graphs which admits a straight-line grid drawing on a grid of area O(n). We give a lineartime algorithm to find such a drawing. Our new class of planar graphs, which we call "doughnut graphs," is a subclass of 5-connected planar graphs. We show several interesting properties of "doughnut graphs" in this paper. One can easily observe that any spanning subgraph of a "doughnut graph" also admits a straight-line grid drawing with linear area. But the recognition of a spanning subgraph of a "doughnut graph" seems to be a non-trivial problem, since the recognition of a spanning subgraph of a given graph is an NP-complete problem in general. We establish a necessary and sufficient condition for a 4-connected planar graph G to be a spanning subgraph of a "doughnut graph." We also give a linear-time algorithm to augment a 4-connected planar graph G to a "doughnut graph" if G satisfies the necessary and sufficient condition.

Discrete Mathematics, 2012

The vertex arboricity va(G) of a graph G is the minimum number of colors the vertices can be labeled so that each color class induces a forest. It was well-known that va(G) ≤ 3 for every planar graph G. In this paper, we prove that va(G) ≤ 2 if G is a planar graph without 7-cycles. This extends a result in [A. Raspaud, W. Wang, On the vertex-arboricity of planar graphs, European J. Combin. 29 (2008) 1064-1075] that for each k ∈ {3, 4, 5, 6}, planar graphs G without k-cycles have va(G) ≤ 2.

Order, 1989

We view the incidence relation of a graph G=(I: E) as an order relation on ~ts vertices and edges, i.e. a <6 b if and only ifa is a vertex and b is an edge incident on a. This leads to the definition of the order-dimension of G as the minimum number of total orders on I'~ E whose intersection is <G. Our main result ~s the characterization of planar graphs as the graphs whose order-dimension does not exceed three. Strong versions of several known properties of planar graphs are ~mphed by this characterizahon These properhes include: each planar graph has arboricn) at most three and each planar graph has a plane embedding ~hose edges are straight line segments. A nice feature of this embedding ~s that the coordinates of the vertices have a purely combmatoriaI meaning.