Related papers: Even and odd trees

The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms…

We determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in P\'olya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity.…

Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in\mathcal{T}$ is…

We consider a uniform spanning tree in a $\delta$-square grid approximation of a planar domain $\Omega$. For given integer $n\ge 2$, we condition the tree on the following $n$-arm event: we pick $n$ branches, emanating from $n$ points…

Any planar graph has a crossing-free straight-line drawing in the plane. A simultaneous geometric embedding of two n-vertex graphs is a straight-line drawing of both graphs on a common set of n points, such that the edges withing each…

This article considers some affine algebraic varieties attached to finite trees and closely related to cluster algebras. Their definition involves a canonical coloring of vertices of trees into three colors. These varieties are proved to be…

A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We…

It is known that every multigraph with an even number of edges has an even orientation (i.e., all indegrees are even). We study parity constrained graph orientations under additional constraints. We consider two types of constraints for a…

In this paper we analyse some questions concerning trees on $\kappa$, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in…

Dual-tree algorithms are a widely used class of branch-and-bound algorithms. Unfortunately, developing dual-tree algorithms for use with different trees and problems is often complex and burdensome. We introduce a four-part logical split:…

Let $ped(n)$ denote the number of partitions of $n$ wherein even parts are distinct (and odd parts are unrestricted). We show infinite families of congruences for $ped(n)$ modulo $8$. We also examine the behavior of $ped_{-2}(n)$ modulo $8$…

We consider an edge-weighted uniform random graph with a given degree sequence (Repeated Configuration Model) which is a useful approximation for many real-world networks. It has been observed that the vertices which are separated from the…

A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a…

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…

We prove that if two graphs of girth at least 6 have isomorphic squares, then the graphs themselves are isomorphic. This is the best possible extension of the results of Ross and Harary on trees and the results of Farzad et al. on graphs of…

Topological phylogenetic trees can be assigned edge weights in several natural ways, highlighting different aspects of the tree. Here the rooted triple and quartet metrizations are introduced, and applied to formulate novel fast methods of…

We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on $n$ vertices,…

For a graph $G$, an edge-separating (resp. vertex-separating) path system of $G$ is a family of paths in $G$ such that for any pair of edges $e_1, e_2$ (resp. pair of vertices $v_1, v_2$) of $G$ there is at least one path in the family that…

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case…

We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with $n$ vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of…