A geometric graph is a graph whose vertices are points in the plane and whose edges are straight-line segments between the points. A plane spanning tree in a geometric graph is a spanning tree that is non-crossing. Let R and B be two disjoint sets of points in the plane where the points of R are colored red and the points of B are colored blue, and let n = |R ∪ B|. A bichromatic plane spanning ...

A plane spanning tree is a tree drawn in the plane so that its edges are closed straight line segments and any two edges do not intersect internally, and no three of its vertices are collinear. In this paper, we present several results on a plane spanning tree such that the graph which is obtained from the tree by adding a line segment between any two end-vertices of the tree is self-intersecting.

The combinatorics of RNA plays a central role in biology. Mathematical biologists have several commonly-used models for RNA: words in a fixed alphabet (representing the primary sequence of nucleotides) and plane trees (representing the secondary structure, or folding of the RNA sequence). This paper considers an augmented version of the standard model of plane trees, one that incorporates some ...

Two plane geometric graphs are said to be compatible when their union is a plane geometric graph. Let S be a set of n points in the Euclidean plane in general position and let T be any given plane geometric tree whose vertex set is S. The main problem we consider in this work consists of finding a second plane geometric tree T ′ on S, such that T ′ is compatible with T and shares with T a minim...

In this paper, we investigate a parking process on uniform random rooted plane tree with $n$ vertices. Every vertex of the has space for single car. Cars arrive at independent uniformly vertices tree. If is unoccupied when car arrives there, it parks. not, drives towards root and parks in first empty encounters (if there one). We are interested asymptotics probability event that all cars can pa...

In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges. 1 Preliminaries This paper is not intended for publication: it does not contain difficult results, and the proofs use only standard and well-known techniques. However, the results stated below have their place in the context of the st...

In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges. 1 Preliminaries This note is not intended for a journal publication: it does not contain difficult results, and the proofs use only standard and well-known techniques. Furthermore, some of the results are already known. However, they ...

If G is a geometric graph with n ≥ 5 vertices and for any set U with 5 vertices of G, the geometric subgraph of G, induced by U , has a plane spanning tree, then G has a plane spanning tree.

The Steiner tree problem requires to nd a shortest tree connecting a given set of terminal points in a metric space. We suggest a better and fast heuristic for the Steiner problem in graphs and in rectilinear plane. This heuristic nds a Steiner tree at most 1.757 and 1.267 times longer than the optimal solution in graphs and rectilinear plane, respectively.