On planar graphs with large tree-width and small grid minors

Journal of Combinatorial Theory, Series B

We prove that every sufficiently large 6-connected graph of bounded treewidth either has a K 6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently large 6-connected graphs. Jørgensen conjectured that it holds for all 6-connected graphs.

Journal of Graph Theory, 2012

It is shown that every sufficiently large almost-5-connected non-planar graph contains a minor isomorphic to an arbitrarily large graph from one of six families of graphs. The graphs in these families are also almost-5-connected, by which we mean that they are 4-connected and all 4-separations contain a "small" side. As a corollary, every sufficiently large almost-5-connected non-planar graph contains both a K 3,4-minor and a K − 6-minor. The connectivity condition cannot be reduced to 4-connectivity, as there are known infinite families of 4-connected non-planar graphs that do not contain a K 3,4-minor. Similarly, there are known infinite families of 4-connected non-planar graphs that do not contain a K − 6-minor.

Theoretical Computer Science, 2011

a r t i c l e i n f o Keywords: Graph algorithms Branch-decompositions Graph minors c α−(c+1) and c β−(2c+1) , respectively.

Journal of Combinatorial Theory, Series B, 2012

At the core of the Robertson-Seymour theory of graph minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing H as a minor. Robertson and Seymour prove several versions of this theorem, each stressing some particular aspects needed at a corresponding stage of the proof of the main result of their theory, the graph minor theorem. We prove a new version of this structure theorem: one that seeks to combine maximum applicability with a minimum of technical ado, and which might serve as a canonical version for future applications in the broader field of graph minor theory. Our proof departs from a simpler version proved explicitly by Robertson and Seymour. It then uses a combination of traditional methods and new techniques to derive some of the more subtle features of other versions as well as further useful properties, with substantially simplified proofs.

Journal of Combinatorial Theory, Series B, 2002

It is shown that for any positive integers k and w there exists a constant N = N (k, w) such that every 7-connected graph of tree-width less than w and of order at least N contains K 3,k as a minor. Similar result is proved for K a,k minors where a is an arbitrary fixed integer and the required connectivity depends only on a. These are the first results of this type where fixed connectivity forces arbitrarily large (nontrivial) minors.

2010

At the core of the Robertson-Seymour theory of graph minors lies a powerful decomposition theorem which captures, for any fixed graph H, the common structural features of all the graphs which do not contain H as a minor. Robertson and Seymour used this result to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation, as well as give a polynomial time algorithm for the disjoint paths problem when the number of the terminals is fixed. The theorem has since found numerous applications, both in graph theory and theoretical computer science. The original proof runs more than 400 pages and the techniques used are highly non-trivial. Robertson and Seymour's proof yields an O(n 3)-time algorithm to find the decomposition. In this paper, we give a simplified algorithm for finding the decomposition based on a new constructive proof of the decomposition theorem for graphs excluding a fixed minor H. The new proof is both dramatically simpler and shorter, making these results and techniques more accessible. The algorithm runs in time O(n 3) as well. Moreover, our proof gives an explicit bound on the constants in the O-notation.

Discrete Applied Mathematics, 2019

We give here some new lower bounds on the order of a largest induced forest in planar graphs with girth 4 and 5. In particular we prove that a triangle-free planar graph of order n admits an induced forest of order at least 6n+7 11 , improving the lower bound of Salavatipour [M. R. Salavatipour, Large induced forests in trianglefree planar graphs, Graphs and Combinatorics, 22:113-126, 2006]. We also prove that a planar graph of order n and girth at least 5 admits an induced forest of order at least 44n+50 69 .

Lecture Notes in Computer Science, 2005

Every lower bound for treewidth can be extended by taking the maximum of the lower bound over all subgraphs or minors. This extension is shown to be a very vital idea for improving treewidth lower bounds. In this paper, we investigate a total of nine graph parameters, providing lower bounds for treewidth. The parameters have in common that they all are the vertex-degree of some vertex in a subgraph or minor of the input graph. We show relations between these graph parameters and study their computational complexity. To allow a practical comparison of the bounds, we developed heuristic algorithms for those parameters that are NP -hard to compute. Computational experiments show that combining the treewidth lower bounds with minors can considerably improve the lower bounds.

Information and Computation, 2011

For several applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast.

2010

For more and more applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast.