As part of the Darwin celebrations in 2009, the Natural History Museum in London unveiled TREE, the first contemporary artwork to win a permanent place in the Museum. While the artist claimed that the inspiration for TREE came from Darwin's famous notebook sketch of branching evolution, sometimes referred to as his "tree of life" drawing, this article emphasises the apparent incongruity between...

Tree decompositions were developed by Robertson and Seymour [21]. Since then algorithms have been developed to solve intractable problems efficiently for graphs of bounded treewidth. In this paper we extend tree decompositions to allow cycles to exist in the decomposition graph; we call these new decompositions plane decompositions because we require that the decomposition graph be planar. Firs...

Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homogeneous metric space such that any real tree with countably many vertices can b...

A minimum Steiner tree for a given set X of points is a network interconnecting the points of X having minimal possible total length. The Steiner ratio for a metric space is the largest lower bound for the ratio of lengths between a minimum Steiner tree and a minimum spanning tree on the same set of points in the metric space. Du et al. (1993) conjectured that the Steiner ratio on a normed plan...

Tree decompositions were developed by Robertson and Seymour [21]. Since then algorithms have been developed to solve intractable problems efficiently for graphs of bounded treewidth. In this paper we extend tree decompositions to allow cycles to exist in the decomposition graph; we call these new decompositions plane decompositions because we require that the decomposition graph be planar. Firs...

In this work we study edge weights for two specific families of increasing trees, which include binary increasing trees and plane oriented recursive trees as special instances, where plane-oriented recursive trees serve as a combinatorial model of scale-free random trees given by the m = 1 case of the BarabásiAlbert model. An edge e = (k, l), connecting the nodes labeled k and l, respectively, ...

When we consider an embedding of graphs into a plane, it would be nice if it does not intersect internally since the embedding simply shows us the structure of graphs. It is easy to embed a tree into a plane with non-self-intersections. If we embed two or more trees into a plane with non-self-intersections, what occurs? In this paper, we consider embeddings of two trees into a plane with using ...

According to the Fibonacci number which is studied by Prodinger et al., we introduce the 2-plane tree which is a planted plane tree with each of its vertices colored with one of two colors and qqppppppppppppppppp -free. The similarity of the enumeration between 2-plane trees and ternary trees leads us to build several bijections. Especially, we found a bijection between the set of 2-plane trees...

Background London’s ‘Major Trauma Network’ was established in April 2010, and comprises four ‘Major Trauma Centres’ (MTCs) – each offering all trauma-related specialties under one roof. Healthcare for London produced an algorithmic ‘Major Trauma Decision Tree’ to help emergency healthcare professionals decide where to convey patients, in order to offer the most suitable care for their needs. Th...

We prove new structural properties for tree-decompositions of planar graphs that we use to improve upon the runtime of tree-decomposition based dynamic programming approaches for several NP-hard planar graph problems. We give for example the fastest algorithm for Planar Dominating Set of runtime 3 · n, when we take the treewidth tw as the measure for the exponential worst case behavior. We also...