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87
Computing Cartograms with Optimal Complexity
"... In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons and edges are represented by sidecontact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a prespecified weight of the corresponding ve ..."
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Cited by 8 (7 self)
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compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8sided rectilinear polygons are necessary, by constructing a nontrivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one
Ramified rectilinear polygons: coordinatization by dendrons
, 2010
"... Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic l1metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectang ..."
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Cited by 5 (4 self)
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. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4cycles or paths of length at most 3
LinearTime Algorithms for Holefree Rectilinear Proportional Contact Graph Representations
"... In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we first study proportional contact representation ..."
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Cited by 2 (1 self)
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representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12sided rectilinear polygons and takes O(n log n) time. We describe a new algorithm that guarantees 10sided
LinearTime Algorithms for Rectilinear Holefree Proportional Contact Representations
"... A proportional contact representation of a planar graph is one where each vertex is represented by a simple polygon with area proportional to a given weight and adjacencies between polygons represent edges between the corresponding pairs of vertices. In this paper we study proportional contact rep ..."
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that produces a holefree proportional contact representation of a maximal planar graph with a 10sided rectilinear polygons. For a planar 3tree we give a lineartime algorithm for a holefree proportional contact representation with 8sided rectilinear polygons. Furthermore, there exist a planar 3tree
The Steiner Tree Problem for Terminals on the Boundary of a Rectilinear Polygon
 In Proc. of the DIMACS Workshop on Network Design: Connectivity and Facilities Location
, 1997
"... Given a simple rectilinear polygon P with k sides and n terminals on its boundary, we present an O(k 3 n)time algorithm to compute the minimal rectilinear Steiner tree lying inside P interconnecting the terminals. We obtain our result by proving structural properties of a selective set of minimal ..."
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Cited by 1 (0 self)
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Given a simple rectilinear polygon P with k sides and n terminals on its boundary, we present an O(k 3 n)time algorithm to compute the minimal rectilinear Steiner tree lying inside P interconnecting the terminals. We obtain our result by proving structural properties of a selective set
On Hamiltonian Triangulations in Simple Polygons (Extended Abstract)
 IN PROCEEDINGS OF THE FIFTH MSISTONY BROOK WORKSHOP ON COMPUTATIONAL GEOMETRY
, 1995
"... An nvertex simple polygon P is said to have a Hamiltonian Triangulation if it has a triangulation whose dual graph contains a Hamiltonian path. Such triangulations are useful in fast rendering engines in Computer Graphics. We give a new characterization of polygons with Hamiltonian triangulations. ..."
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Cited by 8 (1 self)
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An nvertex simple polygon P is said to have a Hamiltonian Triangulation if it has a triangulation whose dual graph contains a Hamiltonian path. Such triangulations are useful in fast rendering engines in Computer Graphics. We give a new characterization of polygons with Hamiltonian triangulations
Segment Endpoint Visibility Graphs are Hamiltonian
 COMPUT. GEOM
, 2002
"... We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian. ..."
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Cited by 10 (3 self)
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We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian.
Graphs with Hamiltonian Balls
"... For a vertex u of a graph G and an integer r, the ball of radius r centered at u is the subgraph Gr(u) induced by the set of all vertices of G whose distance from u does not exceed r. We investigate the set 1i of connected graphs G with at least 3 vertices such that every ball of radius 1 in G has a ..."
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a Hamilton cycle. We prove that every graph G in 1 £ with n vertices has at least 2n 3 edges, and every such graph with 2n 3 edges is isomorphic to a triangulation of a polygon. We show that some wellknown conditions for hamiltonicity of a graph G also guarantee that G has the following property
Hamiltonian Sets of Polygonal Paths in 4Valent Spatial Graphs
, 2012
"... Part of the American Studies Commons, and the Mathematics Commons This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in ..."
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Part of the American Studies Commons, and the Mathematics Commons This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in
Competitive Search Ratio of Graphs and Polygons
 20TH EWCG
, 2004
"... We consider the problem of searching for a goal in an unknown environment, which may be a graph or a polygonal environment. The search ratio is the worstcase ratio before the goal is found while moving along some search path, over the shortest path from the start point to the goal, minimized over a ..."
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the relationship between competitive online exploration and online searching more precisely than before. We apply our framework to searching in trees, (planar) graphs, and in (rectilinear) polygonal environments with or without holes.
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