Results 1 
4 of
4
IMPROVED COMPACT VISIBILITY REPRESENTATION OF Planar Graph via Schnyder’s Realizer
 SIAM J. DISCRETE MATH. C ○ 2004 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS VOL. 18, NO. 1, PP. 19–29
, 2004
"... Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility repre ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easytoimplement O(n)time algorithm bypasses the complicated subroutines for 15 fourconnected components and fourblock trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worstcase lower bound on the required width. Also, if G has no degreethree (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �
AT&T Labs and
"... Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding rectangular layouts is a key problem. We study the areaoptimization problem and show that it is NPhard t ..."
Abstract
 Add to MetaCart
Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding rectangular layouts is a key problem. We study the areaoptimization problem and show that it is NPhard to find a minimumarea rectangular layout of a given contact graph. We present O(n)time algorithms that construct O(n2)area rectangular layouts for general contact graphs and O(n log n)area rectangular layouts for trees. (For trees, this is an O(log n)approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require Ω(n2) (rsp., Ω(n log n)) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of rectangular duals. A corollary to our results relates the class of graphs that admit rectangular layouts to rectangle of influence drawings.