The QuaSEFE Problem Patrizio Angelini, Henry F orster, Michael - - PowerPoint PPT Presentation

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The QuaSEFE Problem Patrizio Angelini, Henry F orster, Michael Hoffmann, Michael Kaufmann, Stephen Kobourov, Giuseppe Liotta, Maurizio Patrignani 27 th International Symposium on Graph Drawing and Network Visualization 2019 The QuaSEFE

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Patrizio Angelini, Henry F¨

  • rster, Michael Hoffmann, Michael Kaufmann,

Stephen Kobourov, Giuseppe Liotta, Maurizio Patrignani

The QuaSEFE Problem

27th International Symposium on Graph Drawing and Network Visualization 2019

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The QuaSEFE Problem

QuaSEFE

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The QuaSEFE Problem

QuaSEFE

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Input: Set of planar graphs with shared vertex set v1 v1 v2 v2 v3 v4 v4 v5 v5 v3

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Input: Set of planar graphs with shared vertex set Output: Planar drawings for all graphs such that v1 v1 v2 v2 v3 v4 v4 v5 v5 v1 v5 v4 v2 v3 v3

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Input: Set of planar graphs with shared vertex set Output: Planar drawings for all graphs such that vertices have the same position in all drawings (simultaneous drawings) v1 v1 v2 v2 v3 v4 v4 v5 v5 v1 v5 v4 v2 v3 v3

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Input: Set of planar graphs with shared vertex set Output: Planar drawings for all graphs such that vertices have the same position in all drawings (simultaneous drawings) edges have the same representation in all drawings (fixed edges) v1 v1 v2 v2 v3 v4 v4 v5 v5 v1 v5 v4 v2 v3 v3

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity Quasiplanar Embedding: No triple of edges crosses pairwise

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity Quasiplanar Embedding: No triple of edges crosses pairwise forbidden

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity Quasiplanar Embedding: No triple of edges crosses pairwise forbidden allowed (no triple)

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity Quasiplanar Embedding: No triple of edges crosses pairwise forbidden allowed (no triple) Thickness two drawings (i.e. two-edge colorable drawings) are quasiplanar

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity QuaSEFE Problem:

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity QuaSEFE Problem: Input: Set of quasiplanar graphs with shared vertex set

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The QuaSEFE Problem

QuaSEFE

Simultaneous (Graph) Embedding with Fixed Edges Quasiplanarity QuaSEFE Problem: Input: Set of quasiplanar graphs with shared vertex set Output: Simultaneous quasiplanar drawings for all graphs with fixed edges

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Related Work

always positive instances for SEFE

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Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06]

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Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06] a planar graph and a tree [Frati ’06]

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Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06] counterexamples for SEFE a planar graph and a tree [Frati ’06]

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Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06] counterexamples for SEFE three paths [Brass et al. ’06] a planar graph and a tree [Frati ’06]

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SLIDE 25

Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06] counterexamples for SEFE three paths [Brass et al. ’06] a planar graph and a tree [Frati ’06] two outerplanar graphs [Frati ’06]

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Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06] counterexamples for SEFE three paths [Brass et al. ’06] a planar graph and a tree [Frati ’06] two outerplanar graphs [Frati ’06] SEFE testable in O(n2) time for two biconnected planar graphs with connected intersection

[Bl¨ asius & Rutter ’16]

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SLIDE 27

Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06] counterexamples for SEFE three paths [Brass et al. ’06] Variants a planar graph and a tree [Frati ’06] two outerplanar graphs [Frati ’06] SEFE testable in O(n2) time for two biconnected planar graphs with connected intersection

[Bl¨ asius & Rutter ’16]

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Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06] counterexamples for SEFE three paths [Brass et al. ’06] Variants no fixed mapping between vertices [Brass et al. ’06] a planar graph and a tree [Frati ’06] two outerplanar graphs [Frati ’06] SEFE testable in O(n2) time for two biconnected planar graphs with connected intersection

[Bl¨ asius & Rutter ’16]

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Related Work

always positive instances for SEFE two caterpillars (in polynomial area) [Brass et al. ’06] counterexamples for SEFE three paths [Brass et al. ’06] Variants no fixed mapping between vertices [Brass et al. ’06] a planar graph and a tree [Frati ’06] two outerplanar graphs [Frati ’06] geometric simultaneous embedding (GSE) [Angelini et al. ’11, Di Giacomo et al. ’15] SEFE testable in O(n2) time for two biconnected planar graphs with connected intersection

[Bl¨ asius & Rutter ’16]

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Related Work - SEFE and Beyond Planarity

quasiplanar GSE

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Related Work - SEFE and Beyond Planarity

quasiplanar GSE a tree and a cycle [Didimo et al. ’12]

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Related Work - SEFE and Beyond Planarity

quasiplanar GSE a tree and a cycle [Didimo et al. ’12] a tree and an outerpillar [Di Giacomo et al. ’15]

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Related Work - SEFE and Beyond Planarity

quasiplanar GSE a tree and a cycle [Didimo et al. ’12] a tree and an outerpillar [Di Giacomo et al. ’15] not every two quasiplanar graphs [Di Giacomo et al. ’15]

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Related Work - SEFE and Beyond Planarity

quasiplanar GSE simultaneous RAC drawings [Argyriou et al. ’13, Bekos et al. ’16, Evans et al. ’16, Grilli ’18] a tree and a cycle [Didimo et al. ’12] a tree and an outerpillar [Di Giacomo et al. ’15] not every two quasiplanar graphs [Di Giacomo et al. ’15]

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Our Results

always positive instances for QuaSEFE

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Our Results

always positive instances for QuaSEFE two planar graphs and a tree

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Our Results

always positive instances for QuaSEFE two planar graphs and a tree a 1-planar graph and a planar graph

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Our Results

always positive instances for QuaSEFE two planar graphs and a tree a 1-planar graph and a planar graph planar graphs with restrictions on their intersection graphs

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Our Results

always positive instances for QuaSEFE two planar graphs and a tree a 1-planar graph and a planar graph planar graphs with restrictions on their intersection graphs counterexamples for QuaSEFE in two special settings

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar
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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 v u1 v u1 u2 u2 u3 u3 u4 u4

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 v u1 v u1 u2 u2 u3 u3 u4 u4 u4

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 v u1 v u1 u2 u2 u3 u3 u4 u4

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 remaining edges embedded planar

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 remaining edges embedded planar

  • 3. Draw G3 quasiplanar
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SLIDE 50

Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 remaining edges embedded planar

  • 3. Draw G3 quasiplanar

G3 \ G1 planar

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SLIDE 51

Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 remaining edges embedded planar

  • 3. Draw G3 quasiplanar

G3 \ G1 planar

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SLIDE 52

Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 remaining edges embedded planar

  • 3. Draw G3 quasiplanar

G3 \ G1 planar

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Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 remaining edges embedded planar

  • 3. Draw G3 quasiplanar

G3 \ G1 planar thickness 2 ⇒ quasiplanar

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SLIDE 54

Two Planar Graphs and a Tree

  • 1. Draw G1 planar

G1 T2 G3

  • 2. Draw T2 planar

some edges fixed by G1 choose planar rotation system from G3 for edges in G3 \ G1 remaining edges embedded planar

  • 3. Draw G3 quasiplanar

G3 \ G1 planar thickness 2 ⇒ quasiplanar

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A 1-Planar Graph and a Planar Graph

  • 1. Decompose the 1-planar graph into a planar graph G1

and a forest T2 [Ackerman ’14]

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A 1-Planar Graph and a Planar Graph

  • 1. Decompose the 1-planar graph into a planar graph G1

and a forest T2 [Ackerman ’14]

  • 2. Apply the previous result (G1 and T2 are planar)
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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V H1 H2 H3 H1,2 H2,3 H H1,3

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H H1,3

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H H1,3

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H H1,3

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H H1,3

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H H1,3

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H Corollary: H1 = ∅ ⇒ QuaSEFE H1,3

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H Corollary: H1 = ∅ ⇒ QuaSEFE H1,3

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H Corollary: H1 = ∅ ⇒ QuaSEFE H1,3 Corollary: H1,2 is forest of paths ⇒ QuaSEFE

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Triples of Planar Graphs

Let G1, G2 and G3 planar graphs on V Theorem: If G1 \ G3, G2 \ G3 admits a SEFE, G1, G2, G3 admits a QuaSEFE. H1 H2 H3 H1,2 H2,3 H Corollary: H1 = ∅ ⇒ QuaSEFE H1,3 Corollary: H1,2 is forest of paths ⇒ QuaSEFE Theorem: If H is a forest of paths, G1, G2, G3 admits a QuaSEFE.

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Sunflower Instances

Sunflower Instance: Planar Graphs G1, . . . , Gk s.t. each edge is either in exactly one Gi or in all Gi (i.e. in H := Gi)

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Sunflower Instances

Sunflower Instance: Planar Graphs G1, . . . , Gk s.t. each edge is either in exactly one Gi or in all Gi (i.e. in H := Gi) Deciding if SEFE exists is NP-hard for k ≥ 3 [Angelini et al. ’15] [Schaefer ’13]

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Sunflower Instances

Sunflower Instance: Planar Graphs G1, . . . , Gk s.t. each edge is either in exactly one Gi or in all Gi (i.e. in H := Gi) Deciding if SEFE exists is NP-hard for k ≥ 3 [Angelini et al. ’15] [Schaefer ’13] Corollary: A sunflower instance with k = 3 planar graphs admits a QuaSEFE.

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Sunflower Instances

Sunflower Instance: Planar Graphs G1, . . . , Gk s.t. each edge is either in exactly one Gi or in all Gi (i.e. in H := Gi) Deciding if SEFE exists is NP-hard for k ≥ 3 [Angelini et al. ’15] [Schaefer ’13] Theorem: For any k, a sunflower instance with k planar graphs admits a QuaSEFE. Corollary: A sunflower instance with k = 3 planar graphs admits a QuaSEFE.

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Sunflower Instances

Sunflower Instance: Planar Graphs G1, . . . , Gk s.t. each edge is either in exactly one Gi or in all Gi (i.e. in H := Gi) Deciding if SEFE exists is NP-hard for k ≥ 3 [Angelini et al. ’15] [Schaefer ’13] Theorem: For any k, a sunflower instance with k planar graphs admits a QuaSEFE.

  • 1. Draw H planar

Corollary: A sunflower instance with k = 3 planar graphs admits a QuaSEFE.

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SLIDE 73

Sunflower Instances

Sunflower Instance: Planar Graphs G1, . . . , Gk s.t. each edge is either in exactly one Gi or in all Gi (i.e. in H := Gi) Deciding if SEFE exists is NP-hard for k ≥ 3 [Angelini et al. ’15] [Schaefer ’13] Theorem: For any k, a sunflower instance with k planar graphs admits a QuaSEFE.

  • 1. Draw H planar
  • 2. Draw each Gi \ H planar

Corollary: A sunflower instance with k = 3 planar graphs admits a QuaSEFE.

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SLIDE 74

Sunflower Instances

Sunflower Instance: Planar Graphs G1, . . . , Gk s.t. each edge is either in exactly one Gi or in all Gi (i.e. in H := Gi) Deciding if SEFE exists is NP-hard for k ≥ 3 [Angelini et al. ’15] [Schaefer ’13] Theorem: For any k, a sunflower instance with k planar graphs admits a QuaSEFE.

  • 1. Draw H planar
  • 2. Draw each Gi \ H planar

each Gi is drawn with thickness 2 Corollary: A sunflower instance with k = 3 planar graphs admits a QuaSEFE.

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Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1.

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Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 v1 v2 v15 v4 v14 v10 v8

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SLIDE 77

Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 M2 = M1 \ {(v17, v18), (v19, v20)} ∪ {(v18, v20)} v1 v2 v15 v4 v14 v10 v8

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SLIDE 78

Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 M2 = M1 \ {(v17, v18), (v19, v20)} ∪ {(v18, v20)} v1 v2 v15 v4 v14 v10 v8

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SLIDE 79

Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 M2 = M1 \ {(v17, v18), (v19, v20)} ∪ {(v18, v20)} v1 v2 v15 v4 v14 v10 v8

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SLIDE 80

Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 M2 = M1 \ {(v17, v18), (v19, v20)} ∪ {(v18, v20)} v1 v2 v15 v4 v14 v10 v8

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SLIDE 81

Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 M2 = M1 \ {(v17, v18), (v19, v20)} ∪ {(v18, v20)} v1 v2 v15 v4 v14 v10 v8

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SLIDE 82

Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 M2 = M1 \ {(v17, v18), (v19, v20)} ∪ {(v18, v20)} v1 v2 v15 v4 v14 v10 v8

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SLIDE 83

Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 M2 = M1 \ {(v17, v18), (v19, v20)} ∪ {(v18, v20)} v1 v2 v15 v4 v14 v10 v8

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SLIDE 84

Drawn Matching and another Matching ✗

Theorem: There are two matchings M1 and M2 that do not admit a QuaSEFE for a fixed drawing of M1. v3 v5 v6 v7 v9 v11 v12 v13 v16 v17 v19 v18 v20 M2 = M1 \ {(v17, v18), (v19, v20)} ∪ {(v18, v20)} v1 v2 v15 v4 v14 v10 v8

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SLIDE 85

Open Problems

Do the following always admit a QuaSEFE?

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SLIDE 86

Open Problems

Do the following always admit a QuaSEFE? two 1-planar graphs

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SLIDE 87

Open Problems

Do the following always admit a QuaSEFE? two 1-planar graphs a quasiplanar graph and a matching

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SLIDE 88

Open Problems

Do the following always admit a QuaSEFE? two 1-planar graphs a quasiplanar graph and a matching three outerplanar graphs

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SLIDE 89

Open Problems

Do the following always admit a QuaSEFE? two 1-planar graphs a quasiplanar graph and a matching three outerplanar graphs four paths

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SLIDE 90

Open Problems

Do the following always admit a QuaSEFE? two 1-planar graphs a quasiplanar graph and a matching three outerplanar graphs four paths What is the computational complexity of QuaSEFE?

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SLIDE 91

Open Problems

Do the following always admit a QuaSEFE? two 1-planar graphs a quasiplanar graph and a matching three outerplanar graphs four paths What is the computational complexity of QuaSEFE? Extend to other beyond planar graph classes such as k-planar graphs.

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SLIDE 92

Open Problems

Do the following always admit a QuaSEFE? two 1-planar graphs a quasiplanar graph and a matching three outerplanar graphs four paths What is the computational complexity of QuaSEFE? Extend to other beyond planar graph classes such as k-planar graphs. Main difficulty: find a similarly catchy name for the problem

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SLIDE 93

Open Problems

Do the following always admit a QuaSEFE? two 1-planar graphs a quasiplanar graph and a matching three outerplanar graphs four paths What is the computational complexity of QuaSEFE? Extend to other beyond planar graph classes such as k-planar graphs. Main difficulty: find a similarly catchy name for the problem

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