Straight-Line Drawings of General Trees with Linear Area and Arbitrary Aspect Ratio

2003 ◽  
pp. 876-885 ◽  
Author(s):  
Ashim Garg ◽  
Adrian Rusu
Keyword(s):  
Aspect Ratio ◽  
Line Drawings ◽  
Straight Line

AREA-EFFICIENT ORDER-PRESERVING PLANAR STRAIGHT-LINE DRAWINGS OF ORDERED TREES

2003 ◽  
Vol 13 (06) ◽  
pp. 487-505 ◽  
Author(s):  
ASHIM GARG ◽  
ADRIAN RUSU
Keyword(s):  
Aspect Ratio ◽  
Binary Tree ◽  
Binary Trees ◽  
Line Drawings ◽  
Ordered Trees ◽  
Grid Drawing ◽  
Aspect Ratios ◽  
Straight Line ◽  
Ordered Tree ◽  
Area Efficient

Ordered trees are generally drawn using order-preserving planar straight-line grid drawings. We investigate the area-requirements of such drawings and present several results. Let T be an ordered tree with n nodes. We show that: • T admits an order-preserving planar straight-line grid drawing with O(n log n) area. • If T is a binary tree, then T admits an order-preserving planar straight-line grid drawing with O(n log log n) area. • If T is a binary tree, then T admits an order-preserving upward planar straight-line grid drawing with optimalO(n log n) area. We also study the problem of drawing binary trees with user-specified aspect ratios. We show that an ordered binary tree T with n nodes admits an order-preserving planar straight-line grid drawing with area O(n log n), and any user-specified aspect ratio in the range [1,n/ log n]. All the drawings mentioned above can be constructed in O(n) time.

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

scholarly journals Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

2017 ◽  
Vol 27 (01n02) ◽  
pp. 121-158 ◽  
Author(s):  
Martin Nöllenburg ◽  
Roman Prutkin ◽  
Ignaz Rutter
Keyword(s):  
Routing Protocol ◽  
Graph Drawing ◽  
Geographic Routing ◽  
Np Hard ◽  
Line Drawings ◽  
Simple Polygons ◽  
Straight Line ◽  
Starting Point ◽  
Minimum Decomposition ◽  
Geographic Routing Protocol

A greedily routable region (GRR) is a closed subset of [Formula: see text], in which any destination point can be reached from any starting point by always moving in the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygonal regions with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles and even for trees, but can be solved optimally for trees in polynomial time, if we allow only certain types of GRR contacts. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.

scholarly journals Distinct Distances in Graph Drawings

10.37236/831 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Paz Carmi ◽  
Vida Dujmović ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Combinatorial Geometry ◽  
Complete Graphs ◽  
Bounded Degree ◽  
Cartesian Products ◽  
Bounded Treewidth ◽  
Line Drawings ◽  
Straight Line ◽  
Minimum Number ◽  
Complete Bipartite Graphs

The distance-number of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in ${\cal O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth $2$ and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree $5$ and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

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