A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to ...

We find a formula to compute the number of the generators, which generate the n-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight n. Applying Hopf algebra of rooted trees, we show that the analogue of Andruskiewitsch and Schneider’s Conjecture is not true. The Hopf algebra of rooted trees and the enveloping algebra of the Lie algeb...

mandibular premolars have earned a reputation for having aberrant anatomy. the occurrence of three canals with three separate foramina in mandibular premolars is very rare. if predictable treatment of a three rooted mandibular premolar is planned, precise knowledge of clinical and radiographic anatomy is absolutely necessary. these teeth may also require special shaping and obturating technique...

a 16 year old female was referred to an oral and maxillofacial radiologist for evaluation of a radiolucent area at mesial radicular aspect of right maxillary central incisor which was detected accidentally during pre-orthodontic treatment check ups. clinical examination showed no particular signs or symptoms except for slightly tenderness to lateral percussion. second radiograph with altered ho...

This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific “transversal structures” on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yi...

We present a new approach in the study of rooted maps without regard to genus. We prove the existence of a new type of equation for the generating series of these maps enumerated with respect to edges and vertices. This is Riccati’s equation. It seems to be the first time that such a differential equation appears in the enumeration of rooted maps. Solving this equation leads to different closed...

Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps have connections, for example, to pattern-avoiding permutations, and they are in one-to-one correspondence with the β(1, 0)-trees introduced by Cori, Jacquard and Schaeffer in 1997. In this paper we enumerate 2-face-free rooted non-separable planar maps and obtain restrict...

The graph copy function when restricted to the set of trees or of rooted trees is called the tree copy function or the rooted tree copy function, respectively. We show that both the set of tree copy functions and the set of rooted tree copy functions are strongly linearly independent. We also show that the set of all tree copy functions are algebraically independent but the set of all rooted tr...

Railway distributed system integration needs to realize information exchange, resources sharing and coordination process across fields, departments and application systems. And railway data integration is essential to implement this integration. In order to resolve the problem of heterogeneity of data models among data sources of different railway operation systems, this paper has brought forth...

Suppose that Gj is a sequence of finite connected planar graphs, and in each Gj a special vertex, called the root, is chosen randomlyuniformly. We introduce the notion of a distributional limit G of such graphs. Assume that the vertex degrees of the vertices in Gj are bounded, and the bound does not depend on j. Then after passing to a subsequence, the limit exists, and is a random rooted graph...