Related papers: On Sets of Lines Not-Supporting Trees
$H$-Packing is the problem of finding a maximum number of vertex-disjoint copies of $H$ in a given graph $G$. $H$-Partition is the special case of finding a set of vertex-disjoint copies that cover each vertex of $G$ exactly once. Our goal…
We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that $(1.293-o(1))n$ points are required to find a straight-line drawing of each $n$-vertex…
In this paper, the problem of pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding patterns of length one and two are obtained. Lagrange inversion formula is…
For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. The enumeration of trees with a…
Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider…
Given a graph $G$ and sets $\{\alpha_v~|~v \in V(G)\}$ and $\{\beta_v~|~v \in V(G)\}$ of non-negative integers, it is known that the decision problem whether $G$ contains a spanning tree $T$ such that $\alpha_v \le d_T (v) \le \beta_v $ for…
We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class…
A tree-based network $\mathcal N$ on $X$ is universal if every rooted binary phylogenetic $X$-tree is a base tree for $\mathcal N$. Hayamizu and, independently, Zhang constructively showed that, for all positive integers $n$, there exists…
We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an…
This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a…
We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that 1. every tree of size $n$ (with arbitrarily large degree) has a straight-line drawing with area $n2^{O(\sqrt{\log\log…
For a forbidden graph $L$, let $ex(p;L)$ denote the maximal number of edges in a simple graph of order $p$ not containing $L$. Let $T_n$ denote the unique tree on $n$ vertices with maximal degree $n-2$, and let $T_n^*=(V,E)$ be the tree on…
Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and…
In this paper, we introduce the Fixed Topology Minimum-Length Tree with Neighborhood Problem, which aims to embed a rooted tree-shaped graph into a $d$-dimensional metric space while minimizing its total length provided that the nodes must…
Given an undirected planar graph G with n vertices and a set S of n points inside a simple polygon P, a point-set embedding of G on S is a planar drawing of G such that each vertex is mapped to a distinct point of S and the edges are…
We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the…
For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem…
Binary jumbled pattern matching asks to preprocess a binary string $S$ in order to answer queries $(i,j)$ which ask for a substring of $S$ that is of length $i$ and has exactly $j$ 1-bits. This problem naturally generalizes to…
We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant…
Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems.…