K 6 minors in 6-connected graphs of bounded tree-width

Discrete Mathematics & Theoretical Computer Science, 2011

We show that for a planar graph with no g-grid minor there exists a tree-decomposition of width at most 5g − 6. The proof is constructive and simple. The underlying algorithm for the tree-decomposition runs in O(n 2 log n) time.

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

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.

Discrete Mathematics, 2008

Thomassen showed in 1994 that all planar graphs are 5-choosable; Škrekovski showed in 1998 that all K 5 -minor-free graphs also are 5-choosable. In this short paper we prove this result using another method.

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.

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.

2006

It is shown that for any positive integer k there exists a constant N = N (k) such that every 7-connected graph of order at least N contains K 3,k as a minor.

Electronic Notes in Discrete Mathematics, 2011

The property that a graph has an embedding in the projective plane is closed under taking minors. Thus by the well known Graph Minor theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it Ω, such that a given graph G is projective planar if and only if G does not contain any graph isomorphic to a member of Ω as a minor. Glover, Huneke and Wang found 35 graphs in Ω, and Archdeacon proved that those are all the members of Ω, but Archdeacon's proof never appeared in any refereed journal. In this thesis we develop a modern approach and technique for finding the list Ω, independent of previous work. Our approach is based on conditioning on the connectivity of a member of Ω. Assume G is a member of Ω. If G is not 3-connected then the structure of G is well understood. In the case that G is 3-connected, the problem breaks down into two main cases, either G has an internal separation of order three or G is internally 4-connected. In this thesis we find the set of all 3-connected minor minimal nonprojective planar graphs with an internal 3-separation. For proving our main result, we use a technique which can be considered as a variation and generalization of the method that Robertson, Seymour and Thomas used for non-planar extension of planar graphs. Using this technique, besides our main result, we also classify the set of minor minimal obstructions for a-, ac-, abc-planarity for rooted graphs. (A rooted graph (G, a, b, c) is a-planar if there exists a split of the vertex a to a ′ and a ′′ in G such that the new graph G ′ obtained by the split has an embedding in a disk such that the vertices a ′ , b, a ′′ , c are on the boundary of the disk in the order listed. We define b-and c-planarity analogously. We say that the rooted graph (G, a, b, c) is ab-planar if it is a-planar or b-planar, and we define abc-planarity analogously.

Journal of Combinatorial Theory, Series B, 2000

In this paper it is shown that any 4-connected graph that does not contain a minor isomorphic to the cube is a minor of the line graph of V n for some n 6 or a minor of one of five graphs. Moreover, there exists a unique 5-connected graph on at least 8 vertices with no cube minor and a unique 4-connected graph with a vertex of degree at least 8 with no cube minor. Further, it is shown that any graph with no cube minor is obtained from 4-connected such graphs by 0-, 1-, and 2-summing, and 3-summing over a specified triangles.

Journal of Combinatorial Theory, Series B, 2002

Let G be a 3-connected planar graph and let U ⊆ V (G). It is shown that G contains a K 2,t minor such that t is large and each vertex of degree 2 in K 2,t corresponds to some vertex of U if and only if there is no small face cover of U. This result cannot be extended to 2-connected planar graphs.