Arithmetic for Rooted Trees

We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication and stretch, prove their properties, and show that all trees can be generated from a starting tree of one vertex. We then show how a given tree can be obtained as the sum or product of two trees, thus defining prime trees with respect to addition and multiplication. In both cases we show how primality can be decided in time polynomial in the number of vertices and we prove that factorization is unique. We then define negative trees and suggest dealing with tree equations, giving some preliminary results. Finally we comment on how our arithmetic might be useful, and discuss preceding studies that have some relations with our. To the best of our knowledge our approach and results are completely new aside for an earlier version of this work submitte as an arXiv manuscript.Comment: 18 pages, 8 figure

Runoff on rooted trees

We introduce an idealised model for overland flow generated by rain falling on a hill-slope. Our prime motivation is to show how the coalescence of runoff streams promotes the total generation of runoff. We show that, for our model, as the rate of rainfall increases in relation to the soil infiltration rate, there is a distinct phase-change. For low rainfall (the subcritical case) only the bottom of the hill-slope contributes to the total overland runoff, while for high rainfall (the supercritical case) the whole slope contributes and the total runoff increases dramatically. We identify the critical point at which the phase-change occurs, and show how it depends on the degree of coalescence. When there is no stream coalescence the critical point occurs when the rainfall rate equals the average infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the average infiltration rate, and increasing the amount of coalescence increases the total expected runoff

On the rooted Tutte polynomial

The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed colors. We establish a number of results pertaining to the rooted Tutte polynomial, including a duality relation in the case that all roots reside around a single face of a planar graph. The connection with the Potts model is also reviewed.Comment: plain latex, 14 pages, 2 figs., to appear in Annales de l'Institut Fourier (1999

On rooted directed path graphs

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. An asteroidal quadruple is a stable set of four vertices such that any three of them is an asteroidal triple. Two non adjacent vertices are linked by a special connection if either they have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain technical conditions. Cameron, Ho`ang, and L´evˆeque [DIMAP Workshop on Algorithmic Graph Theory, 67–74, Electron. Notes Discrete Math., 32, Elsevier, 2009] proved that if a pair of non adjacent vertices are linked by a special connection then in any directed path model T the subpaths of T corresponding to the vertices forming the special connection have to overlap and they force T to be completely directed in one direction between these vertices. Special connections along with the concept of asteroidal quadruple play an important role to study rooted directed path graphs, which are the intersection graphs of directed paths in a rooted directed tree. In this work we define other special connections; these special connections along with the ones defined by Cameron, Ho`ang, and L´evˆeque are nine in total, and we prove that every one forces T to be completely directed in one direction between these vertices. Also, we give a characterization of rooted directed path graphs whose rooted models cannot be rooted on a bold maximal clique. As a by-product of our result, we build new forbidden induced subgraphs for rooted directed path graphs.Fil: Tondato, Silvia Beatriz. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Gutierrez, Marisa. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentin