Simultaneous embeddings with few bends and crossings Fabrizio Frati - PowerPoint PPT Presentation

Simultaneous embeddings with few bends and crossings Fabrizio Frati Michael Hoffmann Vincent Kusters Uni Roma Tre ETH Zurich September 25, 2015

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 .

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 . a b e c d h a b f c g d e

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 . a b a f e c d c e h a b b f c h g d g d e

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 . a b a e c d c e h a b b f c h d g d e

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 . a b a f e c d c e h a b b f c h g d g d e

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 . a b a f e c d c e h a b b f c g d g d e

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 . a b a f e c d c e h a b b f c h g d g d e

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 . a b a f e c d c e h a b b f c h g d g d e

Simultaneous embeddings A simultaneous embedding with fixed edges (SEFE) of G 1 and G 2 is a drawing of G 1 ∪ G 2 that is plane when restricted to G 1 / G 2 . a b a f e c d c e h a b b f c 2 bends h g d g d e

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) .

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE.

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open.

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014:

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014: ▶ 9 bends per edge is always sufficient (Grilli et al. 2014) .

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014: ▶ 9 bends per edge is always sufficient (Grilli et al. 2014) . ▶ 24 crossings per edge pair is always sufficient (Chan et al. 2014) .

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014: ▶ 9 bends per edge is always sufficient (Grilli et al. 2014) . ▶ 24 crossings per edge pair is always sufficient (Chan et al. 2014) . Our paper:

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014: ▶ 9 bends per edge is always sufficient (Grilli et al. 2014) . ▶ 24 crossings per edge pair is always sufficient (Chan et al. 2014) . Our paper: ▶ 1 bend / 4 crossings for tree+tree.

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014: ▶ 9 bends per edge is always sufficient (Grilli et al. 2014) . ▶ 24 crossings per edge pair is always sufficient (Chan et al. 2014) . Our paper: ▶ 1 bend / 4 crossings for tree+tree. ▶ 6 bends / 8 crossings for planar+tree.

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014: ▶ 9 bends per edge is always sufficient (Grilli et al. 2014) . ▶ 24 crossings per edge pair is always sufficient (Chan et al. 2014) . Our paper: ▶ 1 bend / 4 crossings for tree+tree. ▶ 6 bends / 8 crossings for planar+tree. ▶ 6 bends / 16 crossings for planar+planar.

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014: ▶ 9 bends per edge is always sufficient (Grilli et al. 2014) . ▶ 24 crossings per edge pair is always sufficient (Chan et al. 2014) . Our paper: ▶ 1 bend / 4 crossings for tree+tree. (this talk) ▶ 6 bends / 8 crossings for planar+tree. ▶ 6 bends / 16 crossings for planar+planar.

Results Previous work: ▶ Every tree and planar graph admit a SEFE (Frati 2007) . ▶ Some pairs of planar graphs do not admit a SEFE. ▶ Complexity of the decision problem is open. Graph Drawing 2014: ▶ 9 bends per edge is always sufficient (Grilli et al. 2014) . ▶ 24 crossings per edge pair is always sufficient (Chan et al. 2014) . Our paper: ▶ 1 bend / 4 crossings for tree+tree. (this talk) ▶ 6 bends / 8 crossings for planar+tree. (long version) ▶ 6 bends / 16 crossings for planar+planar. (long version)

SEFE of two trees Theorem Let R and B be two trees. There exists a SEFE of R and B in which every edge has at most one bend and every edge pair crosses at most four times.

SEFE of two trees Theorem Let R and B be two trees. There exists a SEFE of R and B in which every edge has at most one bend and every edge pair crosses at most four times. Number of bends is tight (Geyer, Kaufmann, and Vrt’o 2009) .

SEFE of two trees Theorem Let R and B be two trees. There exists a SEFE of R and B in which every edge has at most one bend and every edge pair crosses at most four times. Number of bends is tight (Geyer, Kaufmann, and Vrt’o 2009) . Approach:

SEFE of two trees Theorem Let R and B be two trees. There exists a SEFE of R and B in which every edge has at most one bend and every edge pair crosses at most four times. Number of bends is tight (Geyer, Kaufmann, and Vrt’o 2009) . Approach: 1. Construct combinatorial SEFE of R and B . 2. Contract components of common graph to obtain R ′ and B ′ . 3. Make R ′ and B ′ hamiltonian. 4. Construct SE of R ′ and B ′ with one bend per edge. 5. Expand components of R ′ and B ′ to obtain SEFE of R and B .

1. Construct combinatorial SEFE of R and B . 2. Contract components of common graph to obtain R ′ and B ′ . 3. Make R ′ and B ′ hamiltonian. 4. Construct SE of R ′ and B ′ with one bend per edge. 5. Expand components of R ′ and B ′ to obtain SEFE of R and B .

1. Construct combinatorial SEFE of R and B . 2. Contract components of common graph to obtain R ′ and B ′ . 3. Make R ′ and B ′ hamiltonian. 4. Construct SE of R ′ and B ′ with one bend per edge. 5. Expand components of R ′ and B ′ to obtain SEFE of R and B . S 2 f e d g c h b S 1 a

1. Construct combinatorial SEFE of R and B . 2. Contract components of common graph to obtain R ′ and B ′ . 3. Make R ′ and B ′ hamiltonian. 4. Construct SE of R ′ and B ′ with one bend per edge. 5. Expand components of R ′ and B ′ to obtain SEFE of R and B . S 2 f e d g c h b S 1 a

1. Construct combinatorial SEFE of R and B . 2. Contract components of common graph to obtain R ′ and B ′ . 3. Make R ′ and B ′ hamiltonian. 4. Construct SE of R ′ and B ′ with one bend per edge. 5. Expand components of R ′ and B ′ to obtain SEFE of R and B . S 2 f e d g c h b S 1 a

1. Construct combinatorial SEFE of R and B . 2. Contract components of common graph to obtain R ′ and B ′ . 3. Make R ′ and B ′ hamiltonian. 4. Construct SE of R ′ and B ′ with one bend per edge. 5. Expand components of R ′ and B ′ to obtain SEFE of R and B . S 2 f r ( S 2 ) e d b ( S 2 ) g c b ( S 1 ) h b S 1 r ( S 1 ) a

1. Construct combinatorial SEFE of R and B . 2. Contract components of common graph to obtain R ′ and B ′ . 3. Make R ′ and B ′ hamiltonian. 4. Construct SE of R ′ and B ′ with one bend per edge. 5. Expand components of R ′ and B ′ to obtain SEFE of R and B . S 2 f r ( S 2 ) e d b ( S 2 ) g c b ( S 1 ) h b S 1 r ( S 1 ) a

1. Construct combinatorial SEFE of R and B . 2. Contract components of common graph to obtain R ′ and B ′ . 3. Make R ′ and B ′ hamiltonian. 4. Construct SE of R ′ and B ′ with one bend per edge. 5. Expand components of R ′ and B ′ to obtain SEFE of R and B . S 2 v 2 f f r ( S 2 ) r ( v 2 ) e e d d b ( S 2 ) b ( v 2 ) g g c c b ( S 1 ) b ( v 1 ) h h b b v 1 S 1 r ( S 1 ) r ( v 1 ) a a