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31
Convex Drawings of Graphs in Two and Three Dimensions
, 1996
"... We provide O(n)time algorithms for constructing the following types of drawings of nvertex 3connected planar graphs: ffl 2D convex grid drawings with (3n) × (3n/2) area under the edge L 1 resolution rule; ffl 2D strictly convex grid drawings with O(n³) × O(n³) area under the edge resolution ru ..."
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Cited by 31 (10 self)
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We provide O(n)time algorithms for constructing the following types of drawings of nvertex 3connected planar graphs: ffl 2D convex grid drawings with (3n) × (3n/2) area under the edge L 1 resolution rule; ffl 2D strictly convex grid drawings with O(n³) × O(n³) area under the edge resolution rule; ffl 2D strictly convex drawings with O(1) × O(n) area under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers; ffl 3D convex drawings with O(1) × O(1) × O(n) volume under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers. We also
Rectangular Layouts and Contact Graphs
, 2007
"... 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 ..."
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Cited by 27 (4 self)
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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.
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. ..."
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Cited by 15 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 15 (3 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
Planar Drawings of Plane Graphs
, 2000
"... this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results. ..."
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Cited by 13 (3 self)
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this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.
Really straight graph drawings
 Proc. 12th International Symp. on Graph Drawing (GD ’04
, 2004
"... We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segme ..."
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Cited by 13 (3 self)
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We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of nonplanar graphs with few slopes are also considered. For example, interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the
Octagonal drawings of plane graphs with prescribed face areas
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 2009
"... ..."
Convex Drawings of Graphs in Two and Three Dimensions (Preliminary Version)
"... We provide O(n)time algorithms for constructing the following types of drawings of nvertex 3connected planar graphs: ffl 2D convex grid drawings with (3n) \Theta (3n=2) area under the edge L1 resolution rule; ffl 2D strictly convex grid drawings with O(n 3 ) \Theta O(n 3 ) area under the e ..."
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Cited by 8 (0 self)
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We provide O(n)time algorithms for constructing the following types of drawings of nvertex 3connected planar graphs: ffl 2D convex grid drawings with (3n) \Theta (3n=2) area under the edge L1 resolution rule; ffl 2D strictly convex grid drawings with O(n 3 ) \Theta O(n 3 ) area under the edge resolution rule; ffl 2D strictly convex drawings with O(1) \Theta O(n) area under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers; ffl 3D convex drawings with O(1) \Theta O(1) \Theta O(n) volume under the vertexresolution rule, and with vertex coordinates represented by O(n log n)bit rational numbers. We also show the following lower bounds: ffl For infinitely many nvertex graphs G, if G has a straightline 2D convex drawing in a w \Theta h grid satisfying the edge L1 resolution rule then w;h 5n=6 +\Omega\Gamma20 and w + h 8n=3 + \Omega\Gamma838 ffl For infinitely many boundeddegree triconnected planar graphs G with n ver...
A Simple Linear Time Algorithm for Proper Box Rectangular Drawing of Plane Graphs
 Journal of Algorithms
, 2000
"... In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is dra ..."
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Cited by 6 (0 self)
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In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...
Listing all plane graphs
, 2009
"... In this paper we give a simple algorithm to generate all connected rooted plane graphs with at most m edges. A “rooted” plane graph is a plane graph with one designated (directed) edge on the outer face. The algorithm uses O(m) space and generates such graphs in O(1) time per graph on average withou ..."
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Cited by 5 (1 self)
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In this paper we give a simple algorithm to generate all connected rooted plane graphs with at most m edges. A “rooted” plane graph is a plane graph with one designated (directed) edge on the outer face. The algorithm uses O(m) space and generates such graphs in O(1) time per graph on average without duplications. The algorithm does not output the entire graph but the difference from the previous graph. By modifying the algorithm we can generate all connected (nonrooted) plane graphs with at most m edges in O(m³) time per graph.