The problem of constructing an optimal rooted phylogenetic network from an arbitrary set of rooted triplets is an NP-hard problem. In this paper, we present a heuristic algorithm called TripNet, which tries to construct a rooted phylogenetic network with the minimum number of reticulation nodes from an arbitrary set of rooted triplets. Despite of current methods that work for dense set of roote...

The problem of constructing an optimal rooted phylogenetic‎ ‎network from a set of rooted triplets is NP-hard. ‎In this paper‎, ‎we present a novel method called NCH‎, ‎which tries to construct a rooted phylogenetic network with the minimum number‎‎of reticulation nodes from an arbitrary set of rooted triplets based on the concept of the height function of a tree and a network. We report the pe...

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-dis...

A plane graph is a drawing of a planar graph in the plane such that no two edges cross each other. A rooted plane graph has a designated outer vertex. For given positive integers n ≥ 1 and g ≥ 3, let G3(n, g) denote the set of all triconnected rooted plane graphs with exactly n vertices such that the size of each inner face is at most g. In this paper, we give an algorithm that enumerates all p...

In this paper we give an algorithm to generate all rooted trees with at most n vertices. The algorithm generates each tree in constant time on average. Furthermore the algorithm is simple, and clarifies a simple relation among the trees, that is a family tree of trees, and outputs trees based on the relation.

We describe an algorithm producing circular l a y outs for trees, that is drawings, where subtrees of a node lie within circles, and these circles are themselves placed on the circumference of a circle. The complexity and methodology of our algorithm compares to Reingold and Tilford's algorithm for trees [11]. Moreover, the algorithm naturally admits distortion transformations of the layout. Th...

Suppose that a family of rooted phylogenetic trees Ti with different sets Xi of leaves is given. A supertree for the family is a single rooted tree T whose leaf set is the union of all the Xi, such that the branching information in T corresponds to the branching information in all the trees Ti. This paper proposes a polynomial-time method BUILD-WITH-DISTANCES that makes essential use of distanc...

We give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and proved by Younsi by using the Hurwitz identity on multivariate Abel polynomials. Let [n] = {1, 2, . . . , n}. We also give a bijection between the set of functions from [n +...

For arbitrary positive integers h and m, we consider the family of all rooted trees of height h having exactly m vertices at distance h from the root. We refer to such trees as (h,m)-trees. For a tree T from this family, we consider a simple random walk on T which starts at the root and terminates when it visits one of the m vertices at distance h from the root. Consider the problem of finding ...

Calculations using staggered quarks augmented with a root of the fermion determinant to reduce doubling give a qualitatively incorrect behavior in the small quark mass region. Attempts to circumvent this problem for the continuum limit involve an unproven combination of unphysical states, a loss of unitarity, and a rather peculiar non-commutation of limits.