Packing Plane Spanning Trees and Paths in Complete Geometric Graphs

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph $GK_n$ on any set $S$ of $n$ points in general position in the plane? We show that this number is in $\Omega(\sqrt{n})$. Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).Comment: This work was presented at the 26th Canadian Conference on Computational Geometry (CCCG 2014), Halifax, Nova Scotia, Canada, 2014. The journal version appeared in Information Processing Letters, 124 (2017), 35--4

Edge-Removal and Non-Crossing Configurations in Geometric Graphs

A geometric graph is a graph G = (V;E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V . We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.Postprint (published version

Packing Plane Perfect Matchings into a Point Set

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be packed into any point set $P$. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem

Edge-Removal and Non-Crossing Configurations in Geometric Graphs

Graphs and Algorithm

Packing Non-Self-Crossing Edge-Disjoint Spanning Paths into a Point Set

The term packing refers to the arrangement of multiple geometrical structures or shapes such as circles, squares, triangles, or polygons into a fixed and finite set of points. The geometric structures to be packed can also be trees and paths. Packing is also possible in a 3-dimensional space with geometric structures such as spheres, cylinders, and cubes. The concept of packing was introduced more than half a century ago. Since then, many researchers have studied the packing strategies of different geometric structures in different configurations of point-set. Packing strategies help to construct and arrange multiple geometric structures in a predetermined bounded space; hence, it can be classified as an optimization problem, as we are trying to allocate the optimal space for resources in a finite bounded space. The better the efficiency of the algorithm, the greater number of items that can be packed. Packing geometrical structures have applications in the storage, transportation, and transmission of objects in fields like automobile, aerospace, and naval industries. Since, in real-life scenarios, resources are finite, and space is limited; thus it raises the question, how to efficiently use limited space for accommodating multiple resources. However, packing multiple geometric structures can raise some design considerations. In our research, we have studied the packing of non-self-crossing, edge-disjoint plane spanning paths and have obtained some promising results. We further address some design considerations and provide a different approach on packing at least two non-self-crossing, edge-disjoint plane spanning paths into a point-set