Related papers: Complexity Analysis of Balloon Drawing for Rooted …
A crossing-free straight-line drawing of a graph is monotone if there is a monotone path between any pair of vertices with respect to some direction. We show how to construct a monotone drawing of a tree with $n$ vertices on an $O(n^{1.5})…
The class of self-nested trees presents remarkable compression properties because of the systematic repetition of subtrees in their structure. In this paper, we provide a better combinatorial characterization of this specific family of…
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial…
The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $\mu$ describing the distribution of root hair cells, we seek to maximize a harvest functional $\mathcal{H}$, computing the total…
A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u,w in G, there exists abpath P_{uw} in G that is monotone in some direction l_{uw}. (Namely, the order of the orthogonal projections of…
Measuring the complexity of tree structures can be beneficial in areas that use tree data structures for storage, communication, and processing purposes. This complexity can then be used to compress tree data structures to their…
Geometric embedding of graphs in a point set in the plane is a well known problem. In this paper, the complexity of a variant of this problem, where the point set is bounded by a simple polygon, is considered. Given a point set in the plane…
Computing the rotation distance between two binary trees with $n$ internal nodes efficiently (in $poly(n)$ time) is a long standing open question in the study of height balancing in tree data structures. In this paper, we initiate the study…
Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical…
A gluing of two rooted trees is an identification of their leaves and un-subdivision of the resulting 2-valent vertices. A gluing of two rooted trees is subdivergence free if it has no 2-edge cuts with both roots on the same side of the…
We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to…
This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in set-theorist fashion as partially ordered sets in which the elements below each…
We study the inference of network archaeology in growing random geometric graphs. We consider the root finding problem for a random nearest neighbor tree in dimension $d \in \mathbb{N}$, generated by sequentially embedding vertices…
Fix a sequence c=(c_1,...,c_n) of non-negative integers with sum n-1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v_1,...,v_n so that for each 1 <= i <= n, v_i has exactly c_i children. Let T be…
In this paper, we study how to draw trees so that they are planar, straight-line and respect a given order of edges around each node. We focus on minimizing the height, and show that we can always achieve a height of at most 2pw(T)+1, where…
Frequent tree mining asks us to enumerate tree patterns that occur frequently in a database of rooted trees. This problem is motivated by tree-structured data in bioinformatics, such as glycans and pseudoknot-free RNA secondary structures.…
The early development of a zygote can be mathematically described by a developmental tree. To compare developmental trees of different species, we need to define distances on trees. If children cells after a division are not…
Top-down induction of decision trees has been observed to suffer from the inadequate functioning of the pruning phase. In particular, it is known that the size of the resulting tree grows linearly with the sample size, even though the…
A graph is rectilinear planar if it admits a planar orthogonal drawing without bends. While testing rectilinear planarity is NP-hard in general (Garg and Tamassia, 2001), it is a long-standing open problem to establish a tight upper bound…
Over the past decade, we witness an increasing amount of interest in the design of exact exponential-time and parameterized algorithms for problems in Graph Drawing. Unfortunately, we still lack knowledge of general methods to develop such…