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![Page 1: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/1.jpg)
Upward Planar Morphs
G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli
UNIVERSITÀ DEGLI STUDI
ROMA
TRE
GRAPH DRAWING 2018Barcelona, 26-28 September 2018
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What are we talking about?Morphs
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 3: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/3.jpg)
What are we talking about?Morphs
Transform one drawing into another by moving vertices bypreserving at any time the properties of the drawings.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 4: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/4.jpg)
What are we talking about?Morphs
Transform one drawing into another by moving vertices bypreserving at any time the properties of the drawings.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 5: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/5.jpg)
What are we talking about?Morphs
If vertices move at uniform speed along straight-line trajectories,the morph is linear
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 6: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/6.jpg)
What are we talking about?Morphs
Sometimes linear morphs do not preserve the properties of theinput drawings: some intermediate steps are necessary
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 7: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/7.jpg)
What are we talking about?Morphs
Sometimes linear morphs do not preserve the properties of theinput drawings: some intermediate steps are necessary
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 8: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/8.jpg)
Morphs
Linear MorphsA linear morph is a morph such that vertices move at uniformspeed along straight-line trajectories.
MorphsA morph is a finite sequence of linear morphs, called steps.
The complexity of a morphing algorithm is given by the number ofintermediate morphing steps.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 9: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/9.jpg)
Morphs
Linear MorphsA linear morph is a morph such that vertices move at uniformspeed along straight-line trajectories.
MorphsA morph is a finite sequence of linear morphs, called steps.
The complexity of a morphing algorithm is given by the number ofintermediate morphing steps.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 10: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/10.jpg)
MorphsSome literature
1914-17: Tietze, Smith, Veblen, […] existential proofs for polygons
1944-83: Cairns, Thomassen, […] existential proofs for triangulations and planar graphs
2006: Lubiw et al., SODA morphs of orthogonal drawings in a polynomial number of steps
2013: Alamdari et al., SODA first algorithm for morphing general planar graph drawings in
polynomially many steps2014: Angelini et al., ICALP
optimal (linear) algorithm for planar graphs2014: Barrera-Cruz et al., GD
polynomial algorithm for Schnyder drawings of triangulations2015: Angelini et al., SoCG
optimal (linear) algorithm for convex drawings2018: van Goethem & Verbeek, SoCG
polynomial algorithm for orthogonal drawingsPreliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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What are we talking about?Upward Plane Graphs
• natural extension ofplanarity to directed graphs
• edges drawn upward• vertices with no incoming
edges are sources• vertices with no outgoing
edges are sinks
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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What are we talking about?Upward Plane Graphs
Upward Planar Morphs are defined for upward-equivalent drawings.Upward equivalence is a necessary condition.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Upward DrawingsSome literature
1988: Di Battista & Tamassia drawing algorithms
1992: Di Battista et al. area requirements
1994-98: Bertolazzi et al. drawing algorithms for triconnected, upward planarity
test for single-source2002: Garg & Tamassia
upward planarity test…
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Upward Planar Morphs
Our resultsGiven two upward-equivalent planar straight-line drawings, therealways exists a morph between them such that all the intermediatedrawings of the morph are upward planar and straight-line.
Upward Planar Graphs O(n2) – Ω(n) steps
Reduced Upward Planar Graphs Θ(n) steps
Planar st-Graphs O(n) steps
Reduced Planar st-Graphs Θ(1) steps
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Definitions & Tools
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Left-to-Right equivalence
Two upward-equivalent drawings Γ0 and Γ1 of a graph areLR-equivalent if any horizontal line intersects the same sequence ofvertices and/or edges in both drawings.
For any two LR-equivalent drawings of G, there exists a 1-stepupward planar morph between them.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Bottom-to-Top equivalence
Two upward-equivalent drawings Γ0 and Γ1 of a graph areBT-equivalent if any vertical line intersects the same sequence ofvertices and/or edges in both drawings.
For any two LR-equivalentdrawings of G, there exists a1-step upward planar morphbetween them.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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HVH pairs
Two upward-equivalent planar drawings Γ0 and Γ1 of G are anHVH-pair if there exist Γ′
0 and Γ′1 such that:
• Γ0 and Γ′0 are LR-equivalent
• Γ1 and Γ′1 are LR-equivalent
• Γ′0 and Γ′
1 are BT-equivalent
Γ0
Γ1
LemmaFor any two upward planar drawings that form an HVH pair, thereexists a 3-step upward planar morph between them.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 19: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/19.jpg)
HVH pairs
Two upward-equivalent planar drawings Γ0 and Γ1 of G are anHVH-pair if there exist Γ′
0 and Γ′1 such that:
• Γ0 and Γ′0 are LR-equivalent
• Γ1 and Γ′1 are LR-equivalent
• Γ′0 and Γ′
1 are BT-equivalent
Γ0 Γ′0
Γ1
LemmaFor any two upward planar drawings that form an HVH pair, thereexists a 3-step upward planar morph between them.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 20: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/20.jpg)
HVH pairs
Two upward-equivalent planar drawings Γ0 and Γ1 of G are anHVH-pair if there exist Γ′
0 and Γ′1 such that:
• Γ0 and Γ′0 are LR-equivalent
• Γ1 and Γ′1 are LR-equivalent
• Γ′0 and Γ′
1 are BT-equivalent
Γ0
Γ′1
Γ′0
Γ1
LemmaFor any two upward planar drawings that form an HVH pair, thereexists a 3-step upward planar morph between them.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 21: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/21.jpg)
HVH pairs
Two upward-equivalent planar drawings Γ0 and Γ1 of G are anHVH-pair if there exist Γ′
0 and Γ′1 such that:
• Γ0 and Γ′0 are LR-equivalent
• Γ1 and Γ′1 are LR-equivalent
• Γ′0 and Γ′
1 are BT-equivalent
Γ0
Γ′1
Γ′0
Γ1
LemmaFor any two upward planar drawings that form an HVH pair, thereexists a 3-step upward planar morph between them.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 22: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/22.jpg)
HVH pairs
Two upward-equivalent planar drawings Γ0 and Γ1 of G are anHVH-pair if there exist Γ′
0 and Γ′1 such that:
• Γ0 and Γ′0 are LR-equivalent
• Γ1 and Γ′1 are LR-equivalent
• Γ′0 and Γ′
1 are BT-equivalent
Γ0
Γ′1
Γ′0
Γ1
LemmaFor any two upward planar drawings that form an HVH pair, thereexists a 3-step upward planar morph between them.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 23: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/23.jpg)
Plane st-Graphs
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-Graphs
• G is a DAG with exactly one source andone sink, and they lie on the outer face
• reduced = no transitive edges
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-Graphs
7
• G is a DAG with exactly one source andone sink, and they lie on the outer face
• reduced = no transitive edges
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-Graphs
TheoremFor any two upward-equivalent planar drawings of a reducedst-graph G, there exists a 3-step upward planar morph betweenthem.
Proof: we show that they are an HVH-pair
• assume G biconnected(or augment it)
• exploit anear-decomposition forconstructing Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 27: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/27.jpg)
Reduced Plane st-Graphs
TheoremFor any two upward-equivalent planar drawings of a reducedst-graph G, there exists a 3-step upward planar morph betweenthem.
Proof: we show that they are an HVH-pair
• assume G biconnected(or augment it)
• exploit anear-decomposition forconstructing Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 28: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/28.jpg)
Reduced Plane st-Graphs
TheoremFor any two upward-equivalent planar drawings of a reducedst-graph G, there exists a 3-step upward planar morph betweenthem.
Proof: we show that they are an HVH-pair
• assume G biconnected(or augment it)
• exploit anear-decomposition forconstructing Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
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Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 35: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/35.jpg)
Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 36: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/36.jpg)
Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 37: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/37.jpg)
Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 38: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/38.jpg)
Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 39: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/39.jpg)
Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 40: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/40.jpg)
Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 41: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/41.jpg)
Reduced Plane st-GraphsConstruction of Γ′
0 and Γ′1
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 42: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/42.jpg)
Reduced Plane st-Graphs
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 43: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/43.jpg)
Reduced Plane st-Graphs
TheoremFor any two upward-equivalent planar drawings of an n-vertexplane st-graph G, there exists a O(n)-step upward planar morphbetween them.
Proof
• augment G to be a maximal st-graph, call it G+
• compute upward planar drawings Γ+0 and Γ+
1 of G+
LR-equivalent to Γ0 and Γ1, respectively• contract an internal low-degree vertex v to one of its
neighbors in Γ+0 and Γ+
1 , recursively compute a morph of theobtained drawings, add v back to the obtained morph
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 44: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/44.jpg)
Reduced Plane st-Graphs
TheoremFor any two upward-equivalent planar drawings of an n-vertexplane st-graph G, there exists a O(n)-step upward planar morphbetween them.
Proof• augment G to be a maximal st-graph, call it G+
• compute upward planar drawings Γ+0 and Γ+
1 of G+
LR-equivalent to Γ0 and Γ1, respectively• contract an internal low-degree vertex v to one of its
neighbors in Γ+0 and Γ+
1 , recursively compute a morph of theobtained drawings, add v back to the obtained morph
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 45: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/45.jpg)
Reduced Plane st-Graphs
TheoremFor any two upward-equivalent planar drawings of an n-vertexplane st-graph G, there exists a O(n)-step upward planar morphbetween them.
Proof• augment G to be a maximal st-graph, call it G+
• compute upward planar drawings Γ+0 and Γ+
1 of G+
LR-equivalent to Γ0 and Γ1, respectively
• contract an internal low-degree vertex v to one of itsneighbors in Γ+
0 and Γ+1 , recursively compute a morph of the
obtained drawings, add v back to the obtained morph
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 46: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/46.jpg)
Reduced Plane st-Graphs
TheoremFor any two upward-equivalent planar drawings of an n-vertexplane st-graph G, there exists a O(n)-step upward planar morphbetween them.
Proof• augment G to be a maximal st-graph, call it G+
• compute upward planar drawings Γ+0 and Γ+
1 of G+
LR-equivalent to Γ0 and Γ1, respectively• contract an internal low-degree vertex v to one of its
neighbors in Γ+0 and Γ+
1 , recursively compute a morph of theobtained drawings, add v back to the obtained morph
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 47: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/47.jpg)
Reduced Plane st-GraphsAugmentation to G+ and computation of Γ+
0 and Γ+1
For each internal face f of G:
Use the algorithm for drawing hierarchical plane graphs withassigned y-coordinates by Hong & Nagamochi
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 48: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/48.jpg)
Reduced Plane st-GraphsAugmentation to G+ and computation of Γ+
0 and Γ+1
For each internal face f of G:
Use the algorithm for drawing hierarchical plane graphs withassigned y-coordinates by Hong & Nagamochi
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 49: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/49.jpg)
Reduced Plane st-GraphsAugmentation to G+ and computation of Γ+
0 and Γ+1
For each internal face f of G:
Use the algorithm for drawing hierarchical plane graphs withassigned y-coordinates by Hong & Nagamochi
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 50: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/50.jpg)
Reduced st-GraphsHow to morph maximal plane graphs
Edge contractions are widely used in morphing algorithms
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 51: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/51.jpg)
Reduced st-GraphsHow to morph maximal plane graphs
Edge contractions are widely used in morphing algorithms
lin. morph
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 52: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/52.jpg)
Reduced st-GraphsHow to morph maximal plane graphs
Edge contractions are widely used in morphing algorithms
lin. morph
contr.
contr.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 53: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/53.jpg)
Reduced st-GraphsHow to morph maximal plane graphs
Edge contractions are widely used in morphing algorithms
rec.morph
lin. morph
contr.
contr.
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 54: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/54.jpg)
Reduced Plane st-GraphsDistinguished Neighbors & Contractions
A neighbor u of v is a distinguished neighbor of v if it is:• either a predecessor of v s.t. G has a path from any other
predecessor w to v through u, or• a successor of v s.t. G has a path from v through u to any
other successor w
v v vv
v
v
With a slight variation of the standard technique, we contract v ona distinguished neighbor
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 55: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/55.jpg)
Reduced Plane st-GraphsDistinguished Neighbors & Contractions
A neighbor u of v is a distinguished neighbor of v if it is:• either a predecessor of v s.t. G has a path from any other
predecessor w to v through u, or• a successor of v s.t. G has a path from v through u to any
other successor w
v v vv
v
v
With a slight variation of the standard technique, we contract v ona distinguished neighbor
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 56: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/56.jpg)
Upward Plane Graphs
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 57: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/57.jpg)
(Reduced) Upward Plane Graphs
TheoremFor any two upward equivalent upward planar drawings of ann-vertex upward plane graph G, there exists a O(n2)-step upwardplanar morph between them.
High-level Idea• augment G, Γ0, and Γ1 to obtain a (reduced) st-graph G+
• compute an upward planar morph M between Γ+0 and Γ+
1
• restrict M to the vertices and edges of G
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 58: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/58.jpg)
(Reduced) Upward Plane Graphs
TheoremFor any two upward equivalent upward planar drawings of ann-vertex upward plane graph G, there exists a O(n2)-step upwardplanar morph between them.
High-level Idea• augment G, Γ0, and Γ1 to obtain a (reduced) st-graph G+
• compute an upward planar morph M between Γ+0 and Γ+
1
• restrict M to the vertices and edges of G
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 59: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/59.jpg)
(Reduced) Upward Plane Graphs
TheoremFor any two upward equivalent upward planar drawings of ann-vertex upward plane graph G, there exists a O(n2)-step upwardplanar morph between them.
High-level Idea• augment G, Γ0, and Γ1 to obtain a (reduced) st-graph G+
• compute an upward planar morph M between Γ+0 and Γ+
1
• restrict M to the vertices and edges of G
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 60: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/60.jpg)
(Reduced) Upward Plane Graphs
v
w
u
• G is not an st-graph =⇒ some faceshave more than one source and one sink
• Large angles must occurat some of such vertices
• splitting the face at a Large vertex solvesthe problem locally
• so, add edge uw
• assume that modifying the drawing forinserting uw costs Tuw(n) morphing steps
• G contains O(n) Large vertices
=⇒ turning G into a (reduced) st-graphtakes O(n · Tuw(n)) steps
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 61: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/61.jpg)
(Reduced) Upward Plane Graphs
v
w
u
• G is not an st-graph =⇒ some faceshave more than one source and one sink
• Large angles must occurat some of such vertices
• splitting the face at a Large vertex solvesthe problem locally
• so, add edge uw
• assume that modifying the drawing forinserting uw costs Tuw(n) morphing steps
• G contains O(n) Large vertices=⇒ turning G into a (reduced) st-graph
takes O(n · Tuw(n)) steps
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 62: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/62.jpg)
(Reduced) Upward Plane Graphs
v
w
u
• G is not an st-graph =⇒ some faceshave more than one source and one sink
• Large angles must occurat some of such vertices
• splitting the face at a Large vertex solvesthe problem locally
• so, add edge uw
• assume that modifying the drawing forinserting uw costs Tuw(n) morphing steps
• G contains O(n) Large vertices=⇒ turning G into a (reduced) st-graph
takes O(n · Tuw(n)) steps
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 63: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/63.jpg)
Upward Plane GraphsRemoving Large angles
v
w
u
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 64: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/64.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′v′
v
w
u
insert the gadget
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 65: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/65.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′
v′ v
w
u
triangulate all the faces except those induced by the gadget
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 66: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/66.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′
v′ v
w
uu′
v′′v′
v
w
u
v
w
pvu
u
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 67: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/67.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′v′
v
w
u
insert edges uv′ and uv′′ to obtain an st-graph G∗
where u is a predecessor of both v′ and v′′
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 68: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/68.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′
v′
v
w
u
compute an upward drawing of G∗
…but even by removing v′v′′, u might not have visibility on w
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 69: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/69.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′
v′
v
w
u
compute an upward drawing of G∗
…but even by removing v′v′′, u might not have visibility on w
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 70: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/70.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′
v′
v
w
u
remove edge v′v′′…
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 71: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/71.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′v′
v
w
u
compute a convex upward planar drawing of the obtained st-graphby preserving the y-coordinates (apply Hong & Nagamochi)
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 72: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/72.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′v′
v
w
u
add back v′v′′ and obtain an upward planar drawing of G∗
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 73: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/73.jpg)
Upward Plane GraphsRemoving Large angles
u′
v′′
v′ v
w
uu′
v′′v′
v
w
u
v
w
pvu
u
morph(reduced)st-Graphs
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 74: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/74.jpg)
Upward Planar MorphsConclusions and Open Problems
Upward Planar Graphs O(n2) – Ω(n) steps O(n)?
Reduced Upward Planar Graphs Θ(n) steps
Planar st-Graphs O(n) steps O(1)?
Reduced Planar st-Graphs Θ(1) steps
??-Graphs o(n) steps
Thank you!
Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs
![Page 75: Upward Planar Morphs › gd2018 › wp-content › uploads › 2018 › 11 › s... · 2018-11-05 · Upward Planar Morphs G. Da Lozzo G. Di Battista F. Frati M. Patrignani V. Roselli](https://reader036.vdocument.in/reader036/viewer/2022070818/5f180b54ad978d62da2078ce/html5/thumbnails/75.jpg)
Upward Planar MorphsConclusions and Open Problems
Upward Planar Graphs O(n2) – Ω(n) steps O(n)?
Reduced Upward Planar Graphs Θ(n) steps
Planar st-Graphs O(n) steps O(1)?
Reduced Planar st-Graphs Θ(1) steps
??-Graphs o(n) steps
Thank you!Preliminaries Tools Plane st-Graphs Upward Plane Graphs
G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, V. Roselli Upward Planar Morphs