Schnyder decompositions for regular plane graphs and application to drawing

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to $d$-angulations (plane graphs with faces of degree $d$) for all $d\geq 3$. A \emph{Schnyder decomposition} is a set of $d$ spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly $d-2$ of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the $d$-angulation is $d$. As in the case of Schnyder woods ($d=3$), there are alternative formulations in terms of orientations ("fractional" orientations when $d\geq 5$) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed $d$-angulation of girth $d$ is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on $d$-regular plane graphs of mincut $d$ rooted at a vertex $v^*$) are decompositions into $d$ spanning trees rooted at $v^*$ such that each edge not incident to $v^*$ is used in opposite directions by two trees. Additionally, for even values of $d$, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d=4, these correspond to well-studied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder decompositions yields (planar) orthogonal and straight-line drawing algorithms. For a 4-regular plane graph $G$ of mincut 4 with $n$ vertices plus a marked vertex $v$, the vertices of $G\backslash v$ are placed on a $(n-1) \times (n-1)$ grid according to a permutation pattern, and in the orthogonal drawing each of the $2n-2$ edges of $G\backslash v$ has exactly one bend. Embedding also the marked vertex $v$ is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to $v$. We propose a further compaction step for the drawing algorithm and show that the obtained grid-size is strongly concentrated around $25n/32\times 25n/32$ for a uniformly random instance with $n$ vertices

On the tractability of some natural packing, covering and partitioning problems

In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph $G=(V,E)$ and two "object types" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type $\mathrm{A}$ and one of type $\mathrm{B}$ in the edge set $E$ of $G$, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition $E$ into an object of type $\mathrm{A}$ and one of type $\mathrm{B}$? \textbf{Covering problem:} can we cover $E$ with an object of type $\mathrm{A}$, and an object of type $\mathrm{B}$? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an $s$-$t$ path $P$ and an $s'$-$t'$ path $P'$ that are edge-disjoint. However, many others were not, for example can we find an $s$-$t$ path $P\subseteq E$ and a spanning tree $T\subseteq E$ that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense)

Random incidence matrices: moments of the spectral density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices : any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semi-circle of "small" eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit), we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e=2.72... is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix. Keywords: random graphs, random matrices, sparse matrices, incidence matrices spectrum, momentsComment: 39 pages, 9 figures, Latex2e, [v2: ref. added, Sect. 4 modified

A fast and reliable method for the delineation of tree crown outlines for the computation of crown openness values and other crown parameters

Numerous crown parameters (e.g., leaf area index, diameter, height, volume) can be obtained via the analysis of tree crown photographs. In all cases, parameter values are functions of the position of the crown outline. However, no standardized method to delineate crowns exists. To explore the effect of different outlines on tree crown descriptors, in this case crown openness (CO), and facilitate the adoption of a standard method free of user bias, we developed the program Crown Delineator that automatically delineates any outline around tree crowns following predetermined sensibility settings. We used different outlines to analyze tree CO in contrasting settings: using saplings from four species in young boreal mixedwood forests and medium-sized hybrid poplar trees from a low-density plantation. In both cases, the estimated CO increases when calculated from a looser outline, which had a strong influence on understory available light simulations using a forest simulator. These results demonstrate that the method used to trace crown outlines is an important step in the determination of CO values. We provide a much-needed computer-assisted solution to help standardize this procedure, which can also be used in many other situations in which the delineation of tree crowns is needed (e.g., competition and crown shyness)

The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio

índice de sítio diamétrico: um método alternativo para estimar a qualidade do sítio em florestas de Nothofagus obliqua E N. alpina

The first step for constructing models of tree growth and yield is site quality assessment. To estimate this attribute, several methodologies are available in which site index (SI) is a standard one. However, this approach, that uses height at a reference age of trees, can be simplified if age is replaced by another reference variable easier to measure. In this case, the diametric site index (DSI) represents the mean height of dominant trees at a reference mean diameter at breast height. The aim of this work was to develop DSI in pure and mixed Nothofagus alpina and N. obliqua forests, and compare these models with the classical proposals based on height-age variables, within the temperate forest of northwestern Patagonia from Argentina, South America. Data originated from temporary plots and stem analyses were used. Tree age and diameter at breast height were obtained from each plot and used for establishing DSI family functions, following the guide-curve methodology. Site classes were proportionally represented among DSI curves of 17.0, 21.5, 26.0, 30.5 and 35.0 m of dominant tree height. Reference diameter instead of reference age can be cautiously used in order to fit site index models.Primeiro passo para a construção de modelos de crescimento e produção de árvores e a avaliação da qualidade do sítio. Para estimar este atributo, várias metodologias estão disponíveis, na qual o índice de sítio (IS) é padrão. No entanto, esta abordagem, que utiliza uma altura na idade de referência, pode ser simplificada se a idade é substituída por outra variável de referência mais fácil de medir. Neste caso, o índice de índice de sítio diamétrico (ISD) representa a altura média das árvores dominantes de um diâmetro à altura do peito referência. O objetivo deste trabalho foi desenvolver ISD para florestas puras e mistas de Nothofagus alpina e N. obliqua, e comparar esses modelos com as propostas clássicas baseadas nas variáveis altura-idade, para a floresta temperada do noroeste da Patagônia da Argentina, América do Sul. Dados provenientes de parcelas temporárias e análises de tronco foram utilizados. Foram obtidos idade e diâmetro à altura do peito de cada parcela e utilizados para o estabelecimento das funções da família DSI, seguindo a metodologia da curva-guia. Classes de sítio foram proporcionalmente representados entre curvas DSI de 17,0; 21,5; 26,0; 30,5 e 35,0 m de altura da árvore dominante. O diâmetro de referência em vez da idade de referência pode ser usado com cautela para ajustar modelos de índice de sítio.Fil: Attis Beltran, Hernan. Universidad Nacional del Comahue. Asentamiento Universidad San Martin de Los Andes; Argentina. Universidad Nacional del Comahue; ArgentinaFil: Chauchards, Luis Mario. Universidad Nacional del Comahue; ArgentinaFil: Velásquez, Abel. Universidad Nacional del Comahue; ArgentinaFil: Sbrancia, Renato. Universidad Nacional del Comahue; ArgentinaFil: Martínez Pastur, Guillermo José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Austral de Investigaciones Científicas; Argentina. Universidad Nacional del Comahue; Argentin