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Bound on the faces in a trapezoidal maps

Planar graph, number of facesPlanar graph and number of faces of certain degreeNumber of faces in a planar graph bounded by odd length cycles?Planarity in Graph TheoryUpper bound on the size of the maximum independent setupper bound on the number of edges for a simple planar graphExistence of non-adjacent pair of vertices of small degree in planar graphFind the number of facesNumber of faces in a planar graphFiniteness of regular planar simple graphs with their faces bounded by cycles of the same length

1

$begingroup$

Consider a trapezoidal map used in point location algorithms in computational geometry. Suppose it has $n$ segments. Also, assume that the entire graph is inside an axis parallel to a rectangle. Now, according to the text Computational geometry, Algorithms and applications by Mark de Berg et al. the upper bound on the number of vertices is $6n+4$ . This I get. It's also stated there that the upper bound on the number of faces is $3n+1$ . The proof given there it is not based on Euler's formula for planar graphs, but the authors hint that the bound can also be proved using the said formula by using the bound on the vertices.

Now Euler's formula for a planar graph is f >= 2+e-v. If e and v are not independent, it is not necessary that f reaches maximum when e reaches maximum and v reaches minimum. Even if they are independent, I don't get how upper bound on faces is attained when number of vertices reaches it's maximum. What am I missing here?

graph-theory computational-geometry

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edited 16 hours ago

dmtri

1,5522521

asked 16 hours ago

ajith.mkajith.mk

61

New contributor

ajith.mk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you explain what "a trapezoidal map used in point location algorithms in computational geometry" is, and possibly show an example? (It's not even clear to me from this context whether you mean map in the sense of "a map of England" or map meaning function or something else).
  $endgroup$
  – Henning Makholm
  16 hours ago
  

  
  
  
  


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  $begingroup$
  Consider n 2D line segments. Two line segments are said to be non crossing if their intersection is either empty or one of the end points. All the n line segments are non crossing pairwise. Also no two distinct end points in the set of endpoints of those n segments have same x coordinate. Now enclose all the line segments in an axis parallel rectangle. Draw two vertical lines from each end point. Those verticals stop either at the outer rectangle or when they meet another segment. Create a vertex at each of such intersections. Each face in this planar graph is either a triangle or a trapezoid.
  $endgroup$
  – ajith.mk
  6 mins ago
  

  
  
  
  


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  @HenningMakholm Sorry for the ambiguity. I thought trapezoidal maps are part of the standard terminology in computational geometry.
  $endgroup$
  – ajith.mk
  4 mins ago
  

  
  
  
  

add a comment | 

1

$begingroup$

Consider a trapezoidal map used in point location algorithms in computational geometry. Suppose it has $n$ segments. Also, assume that the entire graph is inside an axis parallel to a rectangle. Now, according to the text Computational geometry, Algorithms and applications by Mark de Berg et al. the upper bound on the number of vertices is $6n+4$ . This I get. It's also stated there that the upper bound on the number of faces is $3n+1$ . The proof given there it is not based on Euler's formula for planar graphs, but the authors hint that the bound can also be proved using the said formula by using the bound on the vertices.

Now Euler's formula for a planar graph is f >= 2+e-v. If e and v are not independent, it is not necessary that f reaches maximum when e reaches maximum and v reaches minimum. Even if they are independent, I don't get how upper bound on faces is attained when number of vertices reaches it's maximum. What am I missing here?

graph-theory computational-geometry

share|cite|improve this question

edited 16 hours ago

dmtri

1,5522521

asked 16 hours ago

ajith.mkajith.mk

61

New contributor

ajith.mk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you explain what "a trapezoidal map used in point location algorithms in computational geometry" is, and possibly show an example? (It's not even clear to me from this context whether you mean map in the sense of "a map of England" or map meaning function or something else).
  $endgroup$
  – Henning Makholm
  16 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Consider n 2D line segments. Two line segments are said to be non crossing if their intersection is either empty or one of the end points. All the n line segments are non crossing pairwise. Also no two distinct end points in the set of endpoints of those n segments have same x coordinate. Now enclose all the line segments in an axis parallel rectangle. Draw two vertical lines from each end point. Those verticals stop either at the outer rectangle or when they meet another segment. Create a vertex at each of such intersections. Each face in this planar graph is either a triangle or a trapezoid.
  $endgroup$
  – ajith.mk
  6 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @HenningMakholm Sorry for the ambiguity. I thought trapezoidal maps are part of the standard terminology in computational geometry.
  $endgroup$
  – ajith.mk
  4 mins ago
  

  
  
  
  

add a comment | 

$begingroup$

Consider a trapezoidal map used in point location algorithms in computational geometry. Suppose it has $n$ segments. Also, assume that the entire graph is inside an axis parallel to a rectangle. Now, according to the text Computational geometry, Algorithms and applications by Mark de Berg et al. the upper bound on the number of vertices is $6n+4$ . This I get. It's also stated there that the upper bound on the number of faces is $3n+1$ . The proof given there it is not based on Euler's formula for planar graphs, but the authors hint that the bound can also be proved using the said formula by using the bound on the vertices.

Now Euler's formula for a planar graph is f >= 2+e-v. If e and v are not independent, it is not necessary that f reaches maximum when e reaches maximum and v reaches minimum. Even if they are independent, I don't get how upper bound on faces is attained when number of vertices reaches it's maximum. What am I missing here?

graph-theory computational-geometry

share|cite|improve this question

edited 16 hours ago

dmtri

1,5522521

asked 16 hours ago

ajith.mkajith.mk

61

New contributor

ajith.mk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited 16 hours ago

dmtri

1,5522521

asked 16 hours ago

ajith.mkajith.mk

61

New contributor

ajith.mk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 16 hours ago

dmtri

1,5522521

asked 16 hours ago

ajith.mkajith.mk

61

New contributor

ajith.mk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited 16 hours ago

dmtri

1,5522521

edited 16 hours ago

dmtri

1,5522521

asked 16 hours ago

ajith.mkajith.mk

61

New contributor

ajith.mk is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 16 hours ago

ajith.mkajith.mk

61