A tree-partition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The tree-partition-width of G is the minimum number of vertices in a bag in a tree-partition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with tree-width k ≥ 3 and maximum degree ∆ ≥ ...

In this paper we present a new width measure for a tree decomposition, minor-matching hypertree width, μ-tw, for graphs and hypergraphs, such that bounding the width guarantees that set of maximal independent sets has a polynomially-sized restriction to each decomposition bag. The relaxed conditions of the decomposition allow a much wider class of graphs and hypergraphs of bounded width compare...

The connected tree-width of a graph is the minimum width of a treedecomposition whose parts induce connected subgraphs. Long cycles are examples of graphs that have small tree-width but large connected tree-width. We show that a graph has small connected tree-width if and only if it has small tree-width and contains no long geodesic cycle. We further prove a connected analogue of the duality th...

The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded treeor pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel...

We prove that the rank-width of an n-vertex graph can be computed exactly in time O(2n log n log log n). To improve over a trivial O(3 log n)-time algorithm, we develop a general framework for decompositions on which an optimal decomposition can be computed efficiently. This framework may be used for other width parameters, including the branch-width of matroids and the carving-width of graphs....

We show that the tree-width of a graph can be defined without reference to graph vertices, and hence the notion of tree-width can be naturally extended to matroids. (This extension was inspired by an original unpublished idea of Jim Geelen.) We prove that the tree-width of a graphic matroid is equal to that of its underlying graph. Furthermore, we extend the well-known relation between the bran...

Multi-clique-width is obtained by a simple modification in the definition of cliquewidth. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the independent set polynomial or testing c-colorabili...

A pplications written for the embedded domain must perform under the constraints of limited memory and limited energy. While these constraints have always existed, current trends, such as mobile computing and ubiquitous computing, bring more and more complex applications to the embedded domain, making performance, or speed of execution, an important factor as well. For instance, we are now able...

−∞ dqD(q). The scattering cross section σ thus approximately decouples into on-shell production (σp) and decay as shown in Eq. (1) for a scalar process [1]. The generalization to multiple resonances is straightforward. We use the notation of [2], Sec. 38. Based on the scales occurring in D(q), the conventional error estimate is O(Γ/M). The branching fraction for the considered decay mode is the...

Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension of a finite poset is bounded in terms of its height and the tree-width of its cover graph.