On the Complexity of HV-Rectilinear Planarity Testing

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On the Complexity of HV-Rectilinear Planarity
Testing
Walter Didimo, Giuseppe Liotta, Maurizio
Patrignani

• Perugia University
• Roma Tre University

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HV-graphs and HV-drawings

• An HV-graph

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• An HV-drawing of it

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HV-rectilinear planarity testing

• A positive instance

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An HV-drawing of it
An HV-graph
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HV-rectilinear planarity testing

• A negative instance

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No HV-drawing can be found
An HV-graph
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Rectilinear drawings

• The HV-rectilinear planarity testing is a
  constrained case of rectilinear planarity testing
• In a rectilinear orthogonal drawing each edge is
  a horizontal or vertical segment and edges do not
  cross

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Rectilinear planarity testing

• Fixed embedding setting
• Polynomial O(n2 log n)
• Tamassia, 87
• Improved to O(n3/2)
• Cornelsen, Karrenbauer, 12
• Linear for maximum degree three
• Rahman, Nishizeki, Naznin, 03
• Variable embedding setting
• NP-complete
• Garg, Tamassia, 01
• Polynomial for biconnected series-parallel graphs
  and 3-planar graphs
• Di Battista, Liotta, Vargiu, 98
• Polynomial for series-parallel graphs of maximum
  degree three
• Zhou, Nishizeki, 08

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Constrained rectilinear planarity

• Left, Right, Up, Down labeling
• Polynomial O(n2)
• Vijayan, Wigderson, 85
• Improved to linear
• Hoffmann, Kriegel, 88
• Polynomial when crossings admitted
• Manuch, Patterson, Poon, Thachuk, 10
• 3D version (Up, Down, Left, Right,
  Front, Back labels)
• Di Battista, Liotta, Lubiw, Whitesides, 02
• Di Giacomo, Liotta, Patrignani, 04
• Di Battista, Kim, Liotta, Lubiw, Whitesides, 12

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HV-rectilinear planarity testing

• Fixed embedding setting
• Polynomial
• Durocher, Felsner, Mehrabi, Mondal, 14
• Variable embedding setting
• Polynomial for biconnected outerplanar graphs
  with vertex-degree at most three
• Durocher, Felsner, Mehrabi, Mondal, 14
• NP-hard when crossings admitted
• Manuch, Patterson, Poon, Thachuk, 10

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HV-rectilinear planarity testing

• Questions
• What is the complexity of HV-rectilinear
  planarity testing in the variable embedding
  setting?
• Manuch, Patterson, Poon, Thachuk, 10
• Durocher, Felsner, Mehrabi, Mondal, 14
• What is the class of HV-outerplanar graphs that
  admit HV-drawings?
• Durocher, Felsner, Mehrabi, Mondal, 14

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Our results

• HV-rectilinear planarity testing is NP-complete
  in the variable embedding setting
• even for HV-graphs with vertex-degree at most
  three
• There exists a polynomial-time algorithm to
  recognize whether a series-parallel HV-graph
  admits an HV-drawing
• extended to partial 2-trees

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Our results

• HV-rectilinear planarity testing is NP-complete
  in the variable embedding setting
• even for HV-graphs with vertex-degree at most
  three
• There exists a polynomial-time algorithm to
  recognize whether a series-parallel HV-graph
  admits an HV-drawing
• extended to partial 2-trees

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Spirality

• Spirality is a measure of how much a path is
  rolled up
• you need spirality 4to close a cycleclockwise

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Switch-flow networks

• A switch-flow network is a graph where each edge
  is labeled with a capacity range c'...c'' of
  nonnegative integers
• For simplicity, the capacity range c...c is
  denoted with c

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Switch-flow networks and flows

• A flow is an orientation of the edges and an
  assignment of integer values to them so that
• Each value is within the capacity range of the
  edge
• The incoming and outgoing flows are balanced at
  each vertex

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Switch-flow network problem

• Instance
• A switch-flow network N
• Question
• Does N admit a flow?
• NP-complete (Garg, Tamassia, 01) even in the
  special case when
• the network is planar
• the lower bounds of the capacity ranges are
  either
• zero (as in 0...c), or
• equal to the upper bounds (as in c)

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1) Start from a switch-flow network
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2) Make it maximal planar

• Dummy edges have all capacity range 0

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3) Compute the dual graph

• The dual graph is 3 regular

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3) Compute the dual graph

• The dual graph is 3 regular
• Capacity ranges are transferred to the edges of
  the dual

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4) Draw the dual orthogonally

• We use Tamassia, Tollis, 89

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5) Transform it into a rigid frame

• Observe that each vertex has maximum degree 3

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Tendrils represents flows

• Let one unit of flow correspond to 4 right angles

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Tendrils

• Tendril Th represents h units of flow between two
  faces

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Wiggles

• A wiggle Wc represents the flow between two faces
  with capacity range 0c

Wiggle W2
Wiggle Wh
Wiggle W1
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6) Produce the final instance
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Equivalence theorem

• A flow in the original network N corresponds to
  an HV-drawing of the constructed instance and
  vice-versa

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Our results

• HV-rectilinear planarity testing is NP-complete
  in the variable embedding setting
• even for HV-graphs with vertex-degree at most
  three
• There exists a polynomial-time algorithm to
  recognize whether a series-parallel HV-graph
  admits an HV-drawing
• extended to partial 2-trees

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What are series-parallel graphs

• A series-parallel graph is either
• A single edge
• The series composition of two series-parallel
  graphs
• The parallel composition of two series-parallel
  graphs
• We consider biconnected series-parallel graphs
• One edge, called reference edge, is in parallel
  with the rest of the graph

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Series-parallel graphs and SPQ-trees

• The decomposition tree describes the series and
  parallel composition needed to build the graph

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Alias vertices and handles

• Detach a component from the graph
• Provide it with suitable handles

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Complex handles
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Replacement theorem

• Given a component, two HV-drawings of it with the
  same spirality are equivalent
• You can replace one with the other and obtain an
  HV-drawing of the graph

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Tuples

• Each component can be decorated with a set of
  O(n) tuples
• each tuple has a value of spirality admitted by
  the component and one realization of such
  spirality

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Algorithm for series-parallel graphs

• Traverse bottom-up the SPQ-tree with reference
  edge e
• For each component compute its set of tuples
  starting from the tuples of its children
• Observe that O(n2) time is sometimes needed
• If one component has zero tuples the instance
  does not admit an HV-drawing with e as the
  reference edge
• Repeat with all possible reference edges

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Main theorem

• There exists an O(n4)-time algorithm that tests
  whether a biconnected series-parallel HV-graph
  with n vertices admits an HV-drawing
• if G has vertex-degree at most 3, the
  time-complexity can be reduced to O(n3 log n).

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Extension to partial 2-trees

• A partial 2-tree is a simply connected graph such
  that each biconnected component is either a
  series-parallel graph or a single edge

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Observation

• Consider two blocks B1 and B2 of the tree that
  have two cut vertices c1 and c2 that can be
  joined by a path not traversing B1 and B2
• In any HV-drawing either c1 is on the external
  face of B1 (B1 is HV-extrovert) or c2 is on the
  external face of B2 (B2 is HV-extrovert)

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Algorithm for partial 2-trees

• Recursively remove HV-extrovert leaf blocks until
  one of the following occurs
• T becomes empty the test is positive
• two blocks that are not HV-extrovert are found
  in this case the test is negative
• T consists of just one block-node B marked as not
  HV-extrovert
• in this case we check whether B admits an
  HV-drawing trying all its edges as reference
  edges

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Open problems

• Can the O(n4) polynomial bound for
  series-parallel graphs be improved?
• for comparison, (unrestricted) rectilinear
  planarity testing of series-parallel graphs with
  vertex-degree at most three is linear
• Zhou, Nishizeki, 08
• Find a combinatorial characterization for the
  HV-graphs that admit an HV-drawing
• e.g. in terms of forbidden substructures

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Thanks!