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17
Transversal structures on triangulations, combinatorial study and straightline drawing
, 2007
"... This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edgelabelling and consists of two transversal bip ..."
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Cited by 25 (5 self)
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This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edgelabelling and consists of two transversal bipolar orientations. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straightline drawing algorithm for irreducible triangulations. For a random irreducible triangulation with n vertices, the grid size of the drawing is asymptotically with high probability 11n/27 × 11n/27 up to an additive error of O ( √ n). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size (⌈n/2 ⌉ − 1) × ⌊n/2⌋.
TriangleFree Planar Graphs as Segment Intersection Graphs
 JOURNAL OF GRAPH ALGORITHMS AND APPLICATIONS
, 2002
"... We prove that every trianglefree planar graph is the intersection graph of a set of segments in the plane. Moreover, the segments can be chosen in only three directions (horizontal, vertical and oblique) and in such a way that no two segments cross, i.e., intersect in a common interior point. Th ..."
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Cited by 18 (0 self)
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We prove that every trianglefree planar graph is the intersection graph of a set of segments in the plane. Moreover, the segments can be chosen in only three directions (horizontal, vertical and oblique) and in such a way that no two segments cross, i.e., intersect in a common interior point. This particular class of intersection graphs is also known as contact graphs.
Transversal structures on triangulations, with application to straight line drawing
 LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We define and study a structure called transversal edgepartition related to triangulations without non empty triangles, which is equivalent to the regular edge labeling discovered by Kant and He. We study other properties of this structure and show that it gives rise to a new straightline drawing ..."
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Cited by 14 (6 self)
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We define and study a structure called transversal edgepartition related to triangulations without non empty triangles, which is equivalent to the regular edge labeling discovered by Kant and He. We study other properties of this structure and show that it gives rise to a new straightline drawing algorithm for triangulations without non empty triangles, and more generally for 4connected plane graphs with at least 4 border vertices. Taking uniformly at random such a triangulation with 4 border vertices and n vertices, the size of the grid is almost surely n
Classes and Recognition of Curve Contact Graphs
"... . Contact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of simple curves, and line segments as a special case, in the plane are considered. Various classes of contact graphs are intro ..."
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Cited by 9 (2 self)
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. Contact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of simple curves, and line segments as a special case, in the plane are considered. Various classes of contact graphs are introduced and the inclusions between them are described, also the recognition of the contact graphs is studied. As one of the main results, it is proved that the recognition of 3contact graphs is NPcomplete for planar graphs, while the same question for planar triangulations is polynomial. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Formally the intersection graph of a set family M is defined as a graph G with the vertex set V (G) = M and the edge set E(G) = \Phi fA; Bg ` M j A 6= B; A " B 6= ; \Psi . Probably the first type studied were interval graphs, see [15],[1]; we may also mention other kinds ...
Combinatorial and Geometric Properties of Planar Laman Graphs
"... Abstract. Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar barandjoint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial struct ..."
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Cited by 7 (3 self)
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Abstract. Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar barandjoint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, and that every angular tree has a corresponding angle labeling and edge labeling. Using a combination of these powerful combinatorial structures, we show that every planar Laman graph has an Lcontact representation, that is, planar Laman graphs are contact graphs of axisaligned Lshapes. Moreover, we show that planar Laman graphs and their subgraphs are the only graphs that can be represented this way. We present efficient algorithms that compute, for every planar Laman graph G, an angular tree, angle labeling, edge labeling, and finally an Lcontact representation of G. The overall running time is O(n 2), where n is the number of vertices of G, and the Lcontact representation is realized on the n × n grid.
Planar graphs as VPGgraphs
, 2013
"... A graph is BkVPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2VPG. We also show that th ..."
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Cited by 7 (3 self)
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A graph is BkVPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2VPG. We also show that the 4connected planar graphs constitute a subclass of the intersection graphs of Zshapes (i.e., a special case of B2VPG). Additionally, we demonstrate that a B2VPG representation of a planar graph can be constructed in O(n 3/2) time. We further show that the trianglefree planar graphs are contact graphs of: Lshapes, Γshapes, vertical segments, and horizontal segments (i.e., a special case of contact B1VPG). From this proof we obtain a new proof that bipartite planar graphs are a subclass of 2DIR.
Orthogonal Segment Stabbing
, 2004
"... We study a class of geometric stabbing/covering problems for sets of line segments, rays, and lines in the plane. While we demonstrate that the problems on sets of horizontal/vertical line segments are NPcomplete, we show that versions involving (parallel) rays or lines are polynomially solvable. ..."
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Cited by 6 (0 self)
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We study a class of geometric stabbing/covering problems for sets of line segments, rays, and lines in the plane. While we demonstrate that the problems on sets of horizontal/vertical line segments are NPcomplete, we show that versions involving (parallel) rays or lines are polynomially solvable.
Planar Graphs Have 1string Representations
, 2010
"... We prove that every planar graph is an intersection graph of strings in the plane such that any two strings intersect at most once. ..."
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Cited by 5 (0 self)
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We prove that every planar graph is an intersection graph of strings in the plane such that any two strings intersect at most once.
Equilateral LContact Graphs
"... Abstract. We consider Lgraphs, that is contact graphs of axisaligned Lshapes in the plane, all with the same rotation. We provide several characterizations of Lgraphs, drawing connections to Schnyder realizers and canonical orders of maximally planar graphs. We show that every contact system of ..."
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Cited by 4 (2 self)
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Abstract. We consider Lgraphs, that is contact graphs of axisaligned Lshapes in the plane, all with the same rotation. We provide several characterizations of Lgraphs, drawing connections to Schnyder realizers and canonical orders of maximally planar graphs. We show that every contact system of L’s can always be converted to an equivalent one with equilateral L’s. This can be used to show a stronger version of a result of Thomassen, namely, that every planar graph can be represented as a contact system of squarebased cuboids. We also study a slightly more restricted version of equilateral Lcontact systems and show that these are equivalent to homothetic triangle contact representations of maximally planar graphs. We believe that this new interpretation of the problem might allow for efficient algorithms to find homothetic triangle contact representations, that do not use Schramm’s monster packing theorem. 1
Contact Graphs of Curves
, 1995
"... Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introdu ..."
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Cited by 4 (4 self)
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Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introduced, their properties and inclusions between them are studied, and the maximal clique in relation with the chromatic number of contact graphs is considered. Also relations between planar and contact graphs are mentioned. Finally, it is proved that the recognition of contact graphs of curves (line segments) is NPcomplete (NPhard) even for planar graphs. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Probably the first type studied were interval graphs (intersection graphs of intervals on a line), owing to their applications in biology, see [14],[1]. We may also mention other kinds of intersection graphs s...