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

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Local and Union Page Numbers

Laura Merker

Karlsruhe Institute of Technology

Torsten Ueckerdt∗

Karlsruhe Institute of Technology

Graph Drawing 2019 September 20, 2019 Pruhonice

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book embedding (≺, P) ⊲ linear vertex ordering ≺ ⊲ edge partition P = {P1, . . . , Pk} ⊲ u ≺ x ≺ v ≺ y, uv ∈ Pi, xy ∈ Pj ⇒ i = j spine ordering pages each page crossing-free K5 K3,3 u v x y

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book embedding (≺, P) ⊲ linear vertex ordering ≺ ⊲ edge partition P = {P1, . . . , Pk} ⊲ u ≺ x ≺ v ≺ y, uv ∈ Pi, xy ∈ Pj ⇒ i = j spine ordering pages each page crossing-free K5 K3,3 u v x y k-local book embedding: each vertex on at most k pages

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book embedding (≺, P) ⊲ linear vertex ordering ≺ ⊲ edge partition P = {P1, . . . , Pk} ⊲ u ≺ x ≺ v ≺ y, uv ∈ Pi, xy ∈ Pj ⇒ i = j spine ordering pages each page crossing-free K5 K3,3 u v x y k-union embedding: each page crossing-free components

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page number pn(G) = min k: ∃ k-page book embedding K3,3 K5 pnℓ pnu pn 2 2 3 2 3 3 each page crossing-free minimize # pages union page number pnu(G) = min k: ∃ k-union embedding local page number pnℓ(G) = min k: ∃ k-local book embedding minimize # pages each page union of crossing-free components each page crossing-free minimize # pages at any one vertex

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G).

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). A simple lower bound . . . |E| =

P ∈P

#{edges in P}

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). A simple lower bound . . . |E| =

P ∈P

#{edges in P} <

P ∈P

2 · #{vertices on P} as each page is outerplanar

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). A simple lower bound . . . |E| =

P ∈P

#{edges in P} <

P ∈P

2 · #{vertices on P} ≤ 2 · pnℓ(G)|V | as each page is outerplanar as each vertex is on at most pnℓ(G) pages

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). A simple lower bound . . . |E| =

P ∈P

#{edges in P} <

P ∈P

2 · #{vertices on P} ≤ 2 · pnℓ(G)|V | Hence pnℓ(G) ≥ |E| 2|V | = 1 4 · avd(G) as each page is outerplanar as each vertex is on at most pnℓ(G) pages avd(G) =

v deg(v)

|V | = 2|E| |V | as each vertex is on at most pnℓ(G) pages

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). A simple lower bound . . . |E| =

P ∈P

#{edges in P} pnℓ(G) ≥ 1 4 mad(G) <

P ∈P

2 · #{vertices on P} ≤ 2 · pnℓ(G)|V | Hence pnℓ(G) ≥ |E| 2|V | = 1 4 · avd(G) = ⇒ as each page is outerplanar as each vertex is on at most pnℓ(G) pages avd(G) =

v deg(v)

|V | = 2|E| |V | as each vertex is on at most pnℓ(G) pages

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

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). . . . gives also an upper bound mad(G) = k = ⇒

rientation with
utdeg(v) ≤ k/2 + 1

A simple lower bound . . . pnℓ(G) ≥ 1 4 mad(G)

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). . . . gives also an upper bound mad(G) = k = ⇒

rientation with
utdeg(v) ≤ k/2 + 1

= ⇒ (k/2 + 2)-local star partition A simple lower bound . . . pnℓ(G) ≥ 1 4 mad(G)

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). . . . gives also an upper bound mad(G) = k = ⇒

rientation with
utdeg(v) ≤ k/2 + 1

= ⇒ (k/2 + 2)-local star partition = ⇒ pnℓ(G) ≤ 1 2 mad(G) + 2 A simple lower bound . . . pnℓ(G) ≥ 1 4 mad(G) as stars are crossing-free

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

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

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Comparison of variants ⊲ For any graph G we have pnℓ(G) ≤ pnu(G) ≤ pn(G). A simple lower bound . . . pnℓ(G) ≥ 1 4 mad(G) . . . gives also an upper bound pnℓ(G) ≤ 1 2 mad(G) + 2 mad(G) = k = ⇒ k + 2 star forests partition = ⇒ pnu(G) ≤ mad(G) + 2 . . . also for union page number Corollary. pnu(G) ≤ 4 pnℓ(G) + 2 but there are n-vertex k-regular graphs with pnu(G) ≤ k + 2 and pn(G) = Ω √ kn

1 2 − 1 k

“local and union page numbers

are tied to density, classical page number is tied to structure”

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Planar graphs pn(G) = 3

max. within

planar graphs pnℓ pnu pn 3 4 2

1 4 mad(G) ≤ pnℓ(G) ≤ pnu(G)

non-hamiltonian triangulation

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Planar graphs pn(G) = 3 pnℓ(G) = pnu(G) = 2

max. within

planar graphs pnℓ pnu pn 3 4 2

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Planar graphs pn(G) = 3 pnℓ(G) = pnu(G) = 2

max. within

planar graphs pnℓ pnu pn 3 4 2 Theorem. pnu(G) ≥ pnℓ(G) ≥ 3 there is a planar graph G with

✗ ✗

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Planar graphs pn(G) = 3 pnℓ(G) = pnu(G) = 2

max. within

planar graphs pnℓ pnu pn 3 4 2

✗ ✗

G planar = ⇒ = ⇒ pnℓ(G) ≤ 4

rientation with
utdeg(v) ≤ 3

= ⇒ 4-local star partition = ⇒ 5 star forest partition = ⇒ pnu(G) ≤ 5 G planar

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k-Trees

(graphs of treewidth k)

1-tree:

r

k-tree:

r

K1 attach to K1 Kk attach to Kk

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k-Trees

(graphs of treewidth k)

1-tree:

r

k-tree:

r

K1 attach to K1 Kk attach to Kk

max. within

k-trees pnℓ pnu pn k k + 1 k/2

1 4 mad(G) ≤ pnℓ(G) ≤ pnu(G)

|E| ≈ k|V | = ⇒ mad(G) ≈ 2k k − 1 . . .

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k-Trees

(graphs of treewidth k)

1-tree:

r

k-tree:

r

K1 attach to K1 Kk attach to Kk

max. within

k-trees pnℓ pnu pn k k + 1 k/2 Theorem. ℓ-local book embedding for every k-tree ℓ-local book embedding for every k-tree with a forest on each page = ⇒

✗ ✗

k − 1

✗ ✗

. . . . . . . . .

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k-Trees

(graphs of treewidth k)

1-tree:

r

k-tree:

r

K1 attach to K1 Kk attach to Kk

max. within

k-trees pnℓ pnu pn k k + 1 k/2

✗ ✗

k − 1

✗ ✗

. . . . . . . . . G k-tree = ⇒ = ⇒ pnℓ(G) ≤ k + 1

rientation with
utdeg(v) ≤ k

= ⇒ (k + 1)-local star partition = ⇒ k + 1 star forest partition = ⇒ pnu(G) ≤ k + 1 G k-tree

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k-Trees

(graphs of treewidth k)

1-tree:

r

k-tree:

r

K1 attach to K1 Kk attach to Kk

max. within

k-trees pnℓ pnu pn k k + 1 k/2

✗ ✗

k − 1

✗ ✗

A possible approach? ⊲ consider the unique (k + 1)-coloring of G ⊲ then any two color classes induce a tree k trees at each vertex can be combined to k or k + 1 forests Still open: Find the spine ordering! . . . . . . . . .

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Complete graphs pnℓ pnu pn ⌈ n

2 ⌉

⌈ n−1

4 ⌉

. . . . . .

n 3

Kn pnℓ(K6) = 2

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Complete graphs pnℓ pnu pn ⌈ n

2 ⌉

⌈ n−1

4 ⌉

. . . . . .

n 3

Kn pnℓ(K6) = 2

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Complete graphs pnℓ pnu pn ⌈ n

2 ⌉

⌈ n−1

4 ⌉

. . . . . .

n 3

Kn pnℓ(K6) = 2

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Complete graphs pnℓ pnu pn ⌈ n

2 ⌉

⌈ n−1

4 ⌉

. . . . . .

n 3

Kn pnℓ(K6) = 2 pnℓ(K9) = 3

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Complete graphs pnℓ pnu pn ⌈ n

2 ⌉

⌈ n−1

4 ⌉

. . . . . .

n 3

Kn pnℓ(K6) = 2 pnℓ(K9) = 3 pnℓ(K11) = 4

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Complete graphs pnℓ pnu pn ⌈ n

2 ⌉

⌈ n−1

4 ⌉

. . . . . .

n 3

Kn pnℓ(K6) = 2 pnℓ(K9) = 3 pnℓ(K11) = 4 pnℓ(K15) ≤ 5

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pnℓ pnu pn ⌈ n

2 ⌉

⌈ n−1

4 ⌉

. . . complete graphs, Kn Open problems pnℓ pnu pn k k + 1 pnℓ pnu pn 3 4 ⊲ computational complexity ? ⊲ maximum pnu(G)/ pnℓ(G) ? ⊲ Km,n ? ⊲ local and union queue numbers ? planar graphs k-trees, treewidth k

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pnℓ pnu pn ⌈ n

2 ⌉

⌈ n−1

4 ⌉

. . . complete graphs, Kn Open problems pnℓ pnu pn k k + 1 pnℓ pnu pn 3 4 ⊲ computational complexity ? ⊲ maximum pnu(G)/ pnℓ(G) ? ⊲ Km,n ? ⊲ local and union queue numbers ? planar graphs k-trees, treewidth k