Large Treewidth Research Articles

On the excluded minor structure theorem for graphs of large tree-width

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.

Polynomial treewidth forces a large grid-like-minor

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ℓ × ℓ grid minor is exponential in ℓ . It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order ℓ in a graph G is a set of paths in G whose intersection graph is bipartite and contains a K ℓ -minor. For example, the rows and columns of the ℓ × ℓ grid are a grid-like-minor of order ℓ + 1 . We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least c ℓ 4 log ℓ has a grid-like-minor of order ℓ . As an application of this result, we prove that the Cartesian product G □ K 2 contains a K ℓ -minor whenever G has treewidth at least c ℓ 4 log ℓ .

Reliability assessment of complex mechatronic systems using a modified nonparametric belief propagation algorithm

Various parametric skewed distributions are widely used to model the time-to-failure (TTF) in the reliability analysis of mechatronic systems, where many items are unobservable due to the high cost of testing. Estimating the parameters of those distributions becomes a challenge. Previous research has failed to consider this problem due to the difficulty of dependency modeling. Recently the methodology of Bayesian networks (BNs) has greatly contributed to the reliability analysis of complex systems. In this paper, the problem of system reliability assessment (SRA) is formulated as a BN considering the parameter uncertainty. As the quantitative specification of BN, a normal distribution representing the stochastic nature of TTF distribution is learned to capture the interactions between the basic items and their output items. The approximation inference of our continuous BN model is performed by a modified version of nonparametric belief propagation (NBP) which can avoid using a junction tree that is inefficient for the mechatronic case because of the large treewidth. After reasoning, we obtain the marginal posterior density of each TTF model parameter. Other information from diverse sources and expert priors can be easily incorporated in this SRA model to achieve more accurate results. Simulation in simple and complex cases of mechatronic systems demonstrates that the posterior of the parameter network fits the data well and the uncertainty passes effectively through our BN based SRA model by using the modified NBP.

Linearity of grid minors in treewidth with applications through bidimensionality

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an ?(w) × ?(w) grid graph as a minor. Thus grid minors suffice to certify that H-minorfree graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms.

Directed Tree-Width

We generalize the concept of tree-width to directed graphs and prove that every directed graph with no “haven” of large order has small tree-width. Conversely, a digraph with a large haven has large tree-width. We also show that the Hamilton cycle problem and other NP-hard problems can be solved in polynomial time when restricted to digraphs of bounded tree-width.

Graph minors. III. Planar tree-width

The “tree-width” of a graph is defined and it is proved that for any fixed planar graph H, every planar graph with sufficiently large tree-width has a minor isomorphic to H. This result has several applications which are described in other papers in this series.