On a Tree and a Path with no Geometric Simultaneous Embedding

Two graphs $G_1=(V,E_1)$ and $G_2=(V,E_2)$ admit a geometric simultaneous embedding if there exists a set of points P and a bijection M: P -> V that induce planar straight-line embeddings both for $G_1$ and for $G_2$. While it is known that two caterpillars always admit a geometric simultaneous embedding and that two trees not always admit one, the question about a tree and a path is still open and is often regarded as the most prominent open problem in this area. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. As a final result, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of depth 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has depth 4.Comment: 42 pages, 33 figure

Superpatterns and Universal Point Sets

An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(n log^O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Strip Planarity Testing of Embedded Planar Graphs

In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph $G(V,E)$ and a function $\gamma:V \rightarrow \{1,2,\dots,k\}$ and asks whether a planar drawing of $G$ exists such that each edge is monotone in the $y$-direction and, for any $u,v\in V$ with $\gamma(u)<\gamma(v)$, it holds $y(u)<y(v)$. The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. We show that the problem is polynomial-time solvable if $G$ has a fixed planar embedding.Comment: 24 pages, 12 figures, extended version of 'Strip Planarity Testing' (21st International Symposium on Graph Drawing, 2013

Three-Dimensional Reconstruction and Modeling Using Low-Precision Vision Sensors for Automation and Robotics Applications in Construction

Automation and robotics in construction (ARC) has the potential to assist in the performance of several mundane, repetitive, or dangerous construction tasks autonomously or under the supervision of human workers, and perform effective site and resource monitoring to stimulate productivity growth and facilitate safety management. When using ARC technologies, three-dimensional (3D) reconstruction is a primary requirement for perceiving and modeling the environment to generate 3D workplace models for various applications. Previous work in ARC has predominantly utilized 3D data captured from high-fidelity and expensive laser scanners for data collection and processing while paying little attention of 3D reconstruction and modeling using low-precision vision sensors, particularly for indoor ARC applications. This dissertation explores 3D reconstruction and modeling for ARC applications using low-precision vision sensors for both outdoor and indoor applications. First, to handle occlusion for cluttered environments, a joint point cloud completion and surface relation inference framework using red-green-blue and depth (RGB-D) sensors (e.g., Microsoft® Kinect) is proposed to obtain complete 3D models and the surface relations. Then, to explore the integration of prior domain knowledge, a user-guided dimensional analysis method using RGB-D sensors is designed to interactively obtain dimensional information for indoor building environments. In order to allow deployed ARC systems to be aware of or monitor humans in the environment, a real-time human tracking method using a single RGB-D sensor is designed to track specific individuals under various illumination conditions in work environments. Finally, this research also investigates the utilization of aerially collected video images for modeling ongoing excavations and automated geotechnical hazards detection and monitoring. The efficacy of the researched methods has been evaluated and validated through several experiments. Specifically, the joint point cloud completion and surface relation inference method is demonstrated to be able to recover all surface connectivity relations, double the point cloud size by adding points of which more than 87% are correct, and thus create high-quality complete 3D models of the work environment. The user-guided dimensional analysis method can provide legitimate user guidance for obtaining dimensions of interest. The average relative errors for the example scenes are less than 7% while the absolute errors less than 36mm. The designed human worker tracking method can successfully track a specific individual in real-time with high detection accuracy. The excavation slope stability monitoring framework allows convenient data collection and efficient data processing for real-time job site monitoring. The designed geotechnical hazard detection and mapping methods enable automated identification of landslides using only aerial video images collected using drones.PHDCivil EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138626/1/yongxiao_1.pd

A Survey of Combinatorial Methods for Phylogenetic Networks

The evolutionary history of a set of species is usually described by a rooted phylogenetic tree. Although it is generally undisputed that bifurcating speciation events and descent with modifications are major forces of evolution, there is a growing belief that reticulate events also have a role to play. Phylogenetic networks provide an alternative to phylogenetic trees and may be more suitable for data sets where evolution involves significant amounts of reticulate events, such as hybridization, horizontal gene transfer, or recombination. In this article, we give an introduction to the topic of phylogenetic networks, very briefly describing the fundamental concepts and summarizing some of the most important combinatorial methods that are available for their computation