Packing trees into planar graphs (original) (raw)

Planar packing of trees and spider trees

Information Processing Letters, 2009

Packing trees into planar graphs, J. Graph Theory (2002) 172-181] García et al. conjectured that for every two non-star trees there exists a planar graph containing them as edge-disjoint subgraphs. In this paper we prove the conjecture in the case in which one of the trees is a spider tree.

Optimal Polygonal Representation of Planar Graphs

Algorithmica, 2012

In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges with slopes 0, 1, -1. Fig. 1. Given a drawing of a planar graph(a), we apportion the edges to the endpoints by cutting each edge in half (b), and then apportion the faces to form polygons (c).

A linear algorithm for compact box-drawings of trees

Networks, 2003

In a box-drawing of a rooted tree, each node is drawn by a rectangular box of prescribed size, no two boxes overlap each other, all boxes corresponding to siblings of the tree have the same x-coordinate at their left sides, and a parent node is drawn at a given distance apart from its first child. A box drawing of a tree is compact if it attains the minimum possible rectangular area enclosing the drawing. We give a linear-time algorithm for finding a compact box-drawing of a tree. A known algorithm does not always find a compact box-drawing and takes time O(n 2) if a tree has n nodes.

Minimum-Area Drawings of Plane 3-Trees (Extended Abstract)

2010

A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The area of such a drawing is the area of the smallest axis-aligned rectangle on the grid which encloses the drawing. A minimum-area drawing of a plane graph G is a straight-line grid drawing of G where the area of the drawing is the minimum. Although it is NP-hard to find minimum-area drawings for general plane graphs, in this paper we obtain minimumarea drawings for plane 3-trees in polynomial time. Furthermore, we show a ⌊2n n 3 − 1 ⌋ × 2⌈ 3 ⌉ lower bound for the area of a straight-line grid drawing of a plane 3-tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1)

Minimum-Area Drawings of Plane 3-Trees

Journal of Graph Algorithms and Applications, 2011

A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The area of such a drawing is the area of the smallest axis-aligned rectangle on the grid which encloses the drawing. A minimum-area drawing of a plane graph G is a straight-line grid drawing of G where the area of the drawing is the minimum. Although it is NP-hard to find minimum-area drawings for general plane graphs, in this paper we obtain minimumarea drawings for plane 3-trees in polynomial time. Furthermore, we show a ⌊ 2n 3 − 1⌋ × 2⌈ n 3 ⌉ lower bound for the area of a straight-line grid drawing of a plane 3tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3 ⌋×⌊ 2(n−1) 3 ⌋ for plane graphs.

Short trees in polygons

Discrete Applied Mathematics, 2002

We consider the problem of determining a short Euclidean tree spanning a number of terminals in a simple polygon. First of all, linear time (in the number of vertices of the polygon) exact algorithms for this problem with three and four terminals are given. Next, these algorithms are used in a fast polynomial heuristic based on the concatenation of trees for appropriately selected subsets with up to four terminals. Computational results indicate that the solutions obtained are close to optimal solutions. ?

Optimal three-dimensional layouts of complete binary trees

Information Processing Letters, 1987

We present optimal embeddings of an n-node complete binary tree in a three-dimensional or a two-dimensional grid when k, the size of one of the dimensions of the grid, is given. For the three-dimensional case we show how to obtain, for any k in the range [1, n/2], a layout of O(n+k logn) volume. The same bound is shown to hold for the two-dimensional case when k is in the range [log n , n/2]. We also show that these bounds are optimal within a constant factor.

Drawing free trees inside simple polygons using polygon skeleton

Computing and Informatics, 2012

Most of graph drawing algorithms draw graphs on unbounded planes. In this paper we introduce a new polyline grid drawing algorithm for drawing free trees on plane regions which are bounded by simple polygons. Our algorithm uses the simulated annealing (SA) ...

Trees with Convex Faces and Optimal Angles

Lecture Notes in Computer Science

We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons. Among all such drawings we seek the one maximizing the angular resolution of the drawing. We find linear time algorithms for solving this problem, both for plane trees and for trees without a fixed embedding. In any such drawing, the edge lengths may be set independently of the angles, without crossing; we describe multiple strategies for setting these lengths.

Complexity of Planar Embeddability of Trees inside Simple Polygons

Corr, 2009

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 bounded by a simple polygon and a free tree, we show that deciding whether there is a planar straight-line embedding of the tree on the point set inside the simple polygon is NP-complete. This implies that the straight-line constrained point-set embedding of trees is also NP-complete, which was posed as an open problem in [8].