Local and Union Page Numbers Torsten Ueckerdt Laura Merker - PowerPoint PPT Presentation

Local and Union Page Numbers Torsten Ueckerdt ∗ Laura Merker Karlsruhe Institute of Technology Karlsruhe Institute of Technology Graph Drawing 2019 September 20, 2019 Pruhonice

book embedding ( ≺ , P ) ⊲ linear vertex ordering ≺ � spine ordering ⊲ edge partition P = { P 1 , . . . , P k } � pages y u x v ⊲ u ≺ x ≺ v ≺ y, uv ∈ P i , xy ∈ P j ⇒ i � = j � each page crossing-free K 5 K 3 , 3

book embedding ( ≺ , P ) ⊲ linear vertex ordering ≺ � spine ordering ⊲ edge partition P = { P 1 , . . . , P k } � pages y u x v ⊲ u ≺ x ≺ v ≺ y, uv ∈ P i , xy ∈ P j ⇒ i � = j � each page crossing-free K 5 K 3 , 3 k -local book embedding: each vertex on at most k pages

book embedding ( ≺ , P ) ⊲ linear vertex ordering ≺ � spine ordering ⊲ edge partition P = { P 1 , . . . , P k } � pages y u x v ⊲ u ≺ x ≺ v ≺ y, uv ∈ P i , xy ∈ P j ⇒ i � = j � each page crossing-free K 5 K 3 , 3 k -union embedding: each page crossing-free components

page number pn( G ) = min k : ∃ k -page book embedding minimize # pages each page crossing-free union page number pn u ( G ) = min k : ∃ k -union embedding minimize # pages each page union of crossing-free components local page number pn ℓ ( G ) = min k : ∃ k -local book embedding minimize # pages each page crossing-free at any one vertex pn ℓ pn u pn K 3 , 3 2 2 3 K 5 2 3 3

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) .

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . � | E | = # { edges in P } P ∈P

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . � | E | = # { edges in P } P ∈P � as each page is outerplanar < 2 · # { vertices on P } P ∈P

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . � | E | = # { edges in P } P ∈P � as each page is outerplanar < 2 · # { vertices on P } P ∈P ≤ 2 · pn ℓ ( G ) | V | as each vertex is on at most pn ℓ ( G ) pages

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . � | E | = # { edges in P } P ∈P � as each page is outerplanar < 2 · # { vertices on P } P ∈P ≤ 2 · pn ℓ ( G ) | V | Hence pn ℓ ( G ) ≥ | E | 2 | V | = 1 as each vertex is on at as each vertex is on at 4 · avd( G ) most pn ℓ ( G ) pages most pn ℓ ( G ) pages � v deg( v ) = 2 | E | avd( G ) = | V | | V |

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . � | E | = # { edges in P } P ∈P � as each page is outerplanar < 2 · # { vertices on P } P ∈P ≤ 2 · pn ℓ ( G ) | V | Hence pn ℓ ( G ) ≥ | E | 2 | V | = 1 as each vertex is on at as each vertex is on at 4 · avd( G ) most pn ℓ ( G ) pages most pn ℓ ( G ) pages pn ℓ ( G ) ≥ 1 4 mad( G ) = ⇒ � v deg( v ) = 2 | E | avd( G ) = | V | | V |

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . pn ℓ ( G ) ≥ 1 4 mad( G ) . . . gives also an upper bound mad( G ) = k

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . pn ℓ ( G ) ≥ 1 4 mad( G ) . . . gives also an upper bound mad( G ) = k orientation with = ⇒ outdeg( v ) ≤ k/ 2 + 1

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . pn ℓ ( G ) ≥ 1 4 mad( G ) . . . gives also an upper bound mad( G ) = k orientation with = ⇒ outdeg( v ) ≤ k/ 2 + 1 = ⇒ ( k/ 2 + 2) -local star partition

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . pn ℓ ( G ) ≥ 1 4 mad( G ) . . . gives also an upper bound mad( G ) = k orientation with = ⇒ outdeg( v ) ≤ k/ 2 + 1 = ⇒ ( k/ 2 + 2) -local star partition ⇒ pn ℓ ( G ) ≤ 1 2 mad( G ) + 2 = as stars are crossing-free

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . pn ℓ ( G ) ≥ 1 4 mad( G ) . . . gives also an upper bound pn ℓ ( G ) ≤ 1 2 mad( G ) + 2

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . pn ℓ ( G ) ≥ 1 4 mad( G ) . . . gives also an upper bound pn ℓ ( G ) ≤ 1 2 mad( G ) + 2 . . . also for union page number mad( G ) = k = ⇒ k + 2 star forests partition = ⇒ pn u ( G ) ≤ mad( G ) + 2

Comparison of variants ⊲ For any graph G we have pn ℓ ( G ) ≤ pn u ( G ) ≤ pn( G ) . A simple lower bound . . . Corollary. pn ℓ ( G ) ≥ 1 4 mad( G ) pn u ( G ) ≤ 4 pn ℓ ( G ) + 2 but there are n -vertex k -regular graphs with . . . gives also an upper bound pn ℓ ( G ) ≤ 1 pn u ( G ) ≤ k + 2 2 mad( G ) + 2 and � √ � 2 − 1 1 pn( G ) = Ω kn k . . . also for union page number mad( G ) = k “local and union page numbers are tied to density, = ⇒ k + 2 star forests partition classical page number is tied to structure” = ⇒ pn u ( G ) ≤ mad( G ) + 2

Planar graphs pn ℓ pn u pn 4 max. within planar graphs 3 2 pn( G ) = 3 1 4 mad( G ) ≤ pn ℓ ( G ) ≤ pn u ( G ) non-hamiltonian triangulation

Planar graphs pn ℓ pn u pn 4 max. within planar graphs 3 2 pn( G ) = 3 pn ℓ ( G ) = pn u ( G ) = 2

Planar graphs pn ℓ pn u pn 4 max. within planar graphs 3 ✗ ✗ 2 pn( G ) = 3 pn ℓ ( G ) = pn u ( G ) = 2 Theorem. there is a planar graph G with pn u ( G ) ≥ pn ℓ ( G ) ≥ 3

Planar graphs pn ℓ pn u pn 4 max. within planar graphs 3 ✗ ✗ 2 G planar pn( G ) = 3 orientation with = ⇒ outdeg( v ) ≤ 3 pn ℓ ( G ) = pn u ( G ) = 2 = ⇒ 4 -local star partition = ⇒ pn ℓ ( G ) ≤ 4 G planar = ⇒ 5 star forest partition = ⇒ pn u ( G ) ≤ 5

k -Trees (graphs of treewidth k ) 1 -tree: or K 1 attach to K 1 k -tree: or K k attach to K k

k -Trees (graphs of treewidth k ) 1 -tree: pn ℓ pn u pn or k + 1 max. within K 1 attach to K 1 k -trees k k -tree: k − 1 . . . or k/ 2 K k attach to K k 1 4 mad( G ) ≤ pn ℓ ( G ) ≤ pn u ( G ) | E | ≈ k | V | = ⇒ mad( G ) ≈ 2 k

k -Trees (graphs of treewidth k ) 1 -tree: pn ℓ pn u pn or k + 1 max. within K 1 attach to K 1 k -trees k k -tree: ✗ ✗ k − 1 . . . . . . . . . or ✗ ✗ k/ 2 K k attach to K k Theorem. ℓ -local book embedding ℓ -local book embedding for every = ⇒ for every k -tree k -tree with a forest on each page

k -Trees (graphs of treewidth k ) 1 -tree: pn ℓ pn u pn or k + 1 max. within K 1 attach to K 1 k -trees k k -tree: ✗ ✗ k − 1 . . . . . . . . . or ✗ ✗ k/ 2 K k attach to K k G k -tree G k -tree orientation with = ⇒ k + 1 star forest partition = ⇒ outdeg( v ) ≤ k = ⇒ pn u ( G ) ≤ k + 1 = ⇒ ( k + 1) -local star partition = ⇒ pn ℓ ( G ) ≤ k + 1

k -Trees (graphs of treewidth k ) 1 -tree: pn ℓ pn u pn or k + 1 max. within K 1 attach to K 1 k -trees k k -tree: ✗ ✗ k − 1 . . . . . . . . . or ✗ ✗ k/ 2 K k attach to K k A possible approach? ⊲ consider the unique ( k + 1) -coloring of G ⊲ then any two color classes induce a tree Still open: � k trees at each vertex Find the spine ordering! � can be combined to k or k + 1 forests

Complete graphs pn ℓ pn u pn ⌈ n 2 ⌉ K n . . . n 3 . . . pn ℓ ( K 6 ) = 2 ⌈ n − 1 4 ⌉

Complete graphs pn ℓ pn u pn ⌈ n 2 ⌉ K n . . . n 3 . . . pn ℓ ( K 6 ) = 2 ⌈ n − 1 4 ⌉

Complete graphs pn ℓ pn u pn ⌈ n 2 ⌉ K n . . . n 3 . . . pn ℓ ( K 6 ) = 2 ⌈ n − 1 4 ⌉

Complete graphs pn ℓ pn u pn ⌈ n 2 ⌉ K n . . . n 3 . . . pn ℓ ( K 6 ) = 2 ⌈ n − 1 4 ⌉ pn ℓ ( K 9 ) = 3

Complete graphs pn ℓ pn u pn ⌈ n 2 ⌉ K n . . . n 3 . . . pn ℓ ( K 6 ) = 2 ⌈ n − 1 4 ⌉ pn ℓ ( K 9 ) = 3 pn ℓ ( K 11 ) = 4

Complete graphs pn ℓ pn u pn ⌈ n 2 ⌉ K n . . . n 3 . . . pn ℓ ( K 6 ) = 2 ⌈ n − 1 4 ⌉ pn ℓ ( K 15 ) ≤ 5 pn ℓ ( K 9 ) = 3 pn ℓ ( K 11 ) = 4

Open problems pn ℓ pn u pn pn ℓ pn u pn ⌈ n 2 ⌉ k + 1 . . . k ⌈ n − 1 4 ⌉ k -trees, treewidth k complete graphs, K n pn ℓ pn u pn 4 3 planar graphs ⊲ computational complexity ? ⊲ K m,n ? ⊲ maximum pn u ( G ) / pn ℓ ( G ) ? ⊲ local and union queue numbers ?

Open problems pn ℓ pn u pn pn ℓ pn u pn ⌈ n 2 ⌉ k + 1 . . . k ⌈ n − 1 4 ⌉ k -trees, treewidth k complete graphs, K n pn ℓ pn u pn 4 Thank you 3 planar graphs ⊲ computational complexity ? ⊲ K m,n ? ⊲ maximum pn u ( G ) / pn ℓ ( G ) ? ⊲ local and union queue numbers ?