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Towards characterizing graphs with a sliceable rectangular dual Vincent Kusters Bettina Speckmann ETH Zurich TU Eindhoven September 26, 2015 Cartograms Cartograms NO IS SE FI DK EE RU LT LV IE GB BY PL NL DE BE UA CZ SK

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Towards characterizing graphs with a sliceable rectangular dual

Vincent Kusters Bettina Speckmann

ETH Zurich TU Eindhoven

September 26, 2015

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Cartograms

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Cartograms

DE BE UA CY MT MK BA EE NO GR AL RU CH IE PL RO SE CS HU CZ IS PT TR IT ES BG GB LT DK BY SI LV SK NL FI AT FR HR

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Rectangular partitions

A rectangular partition of a rectangle R is a set of non-overlapping rectangles that together form R.

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Rectangular partitions

A rectangular partition of a rectangle R is a set of non-overlapping rectangles that together form R.

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Rectangular partitions

A rectangular partition of a rectangle R is a set of non-overlapping rectangles that together form R.

✔

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Rectangular partitions

A rectangular partition of a rectangle R is a set of non-overlapping rectangles that together form R.

✔ ✗

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Rectangular duals

A rectangular dual of a planar graph G is a rectangular partition R, such that:

▶ vertices in G correspond to rectangles in R and ▶ edges in G correspond to shared borders in R.

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SLIDE 9

Rectangular duals

A rectangular dual of a planar graph G is a rectangular partition R, such that:

▶ vertices in G correspond to rectangles in R and ▶ edges in G correspond to shared borders in R.

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SLIDE 10

Corner assignments

A corner assignment or extended graph E(G) of G is an extention of G with four vertices. The four vertices form the outer cycle of E(G).

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Corner assignments

A corner assignment or extended graph E(G) of G is an extention of G with four vertices. The four vertices form the outer cycle of E(G).

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Corner assignments

A corner assignment or extended graph E(G) of G is an extention of G with four vertices. The four vertices form the outer cycle of E(G).

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Regular edge labelings

A regular edge labeling of an extended graph E(G) is a partition of the interior edges of E(G) into red and blue directed edges.

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Regular edge labelings

A regular edge labeling of an extended graph E(G) is a partition of the interior edges of E(G) into red and blue directed edges.

. . .

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Regular edge labelings

A regular edge labeling of an extended graph E(G) is a partition of the interior edges of E(G) into red and blue directed edges.

. . . . . .

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SLIDE 16

Regular edge labelings

A regular edge labeling of an extended graph E(G) is a partition of the interior edges of E(G) into red and blue directed edges.

. . . . . . . . .

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SLIDE 17

Regular edge labelings

A regular edge labeling of an extended graph E(G) is a partition of the interior edges of E(G) into red and blue directed edges.

. . . . . . . . . . . .

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SLIDE 18

Regular edge labelings

A regular edge labeling of an extended graph E(G) is a partition of the interior edges of E(G) into red and blue directed edges.

. . . . . . . . . . . .

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SLIDE 19

Regular edge labelings

A regular edge labeling of an extended graph E(G) is a partition of the interior edges of E(G) into red and blue directed edges.

. . . . . . . . . . . .

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Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 23

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 24

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 25

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 26

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 27

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 32

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 33

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 34

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 35

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 36

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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SLIDE 37

Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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Sliceable rectangular duals

A rectangular dual is sliceable if it can be recursively partitioned along horizontal or vertical lines:

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

then not sliceable (Ko´

zmi´ nski and Kinnen 1985)

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

then not sliceable (Ko´

zmi´ nski and Kinnen 1985)

▶ G has no separating 4-cycle

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

then not sliceable (Ko´

zmi´ nski and Kinnen 1985)

▶ G has no separating 4-cycle

then sliceable (Yeap and Sarrafzadeh 1995)

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

then not sliceable (Ko´

zmi´ nski and Kinnen 1985)

▶ G has no separating 4-cycle

then sliceable (Yeap and Sarrafzadeh 1995)

▶ G has a separating 4-cycle

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

then not sliceable (Ko´

zmi´ nski and Kinnen 1985)

▶ G has no separating 4-cycle

then sliceable (Yeap and Sarrafzadeh 1995)

▶ G has a separating 4-cycle

then ???

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SLIDE 47

Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

then not sliceable (Ko´

zmi´ nski and Kinnen 1985)

▶ G has no separating 4-cycle

then sliceable (Yeap and Sarrafzadeh 1995)

▶ G has a separating 4-cycle

then ???

▶ G has exactly one separating 4-cycle

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SLIDE 48

Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ E(G) has a separating 3-cycle

then not sliceable (Ko´

zmi´ nski and Kinnen 1985)

▶ G has no separating 4-cycle

then sliceable (Yeap and Sarrafzadeh 1995)

▶ G has a separating 4-cycle

then ???

▶ G has exactly one separating 4-cycle

then sliceable ⟺ not rotating windmill

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ G has exactly one separating 4-cycle

then sliceable ⟺ not rotating windmill

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ G has exactly one separating 4-cycle

then sliceable ⟺ not rotating windmill

Proof.

⟹: Show that rotating windmills are not sliceable.

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Characterizing sliceable graphs

An extended graph is sliceable if it has a sliceable rectangular dual.

▶ G has exactly one separating 4-cycle

then sliceable ⟺ not rotating windmill

Proof.

⟹: Show that rotating windmills are not sliceable. ⟸: Given an extended graph E(G) that is not a rotating windmill, show that we can find a slice that splits E(G) into extended graphs that are not rotating windmills.

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Rotating windmills

1 4 ↑ 3

The following extended graphs are rotating windmills:

▶ the windmill, ▶ the four base rotating windmills, ▶ any extended graph obtained by applying one of

three construction steps.

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Rotating windmills

1 4 ↑ 3

The following extended graphs are rotating windmills:

▶ the windmill, ▶ the four base rotating windmills, ▶ any extended graph obtained by applying one of

three construction steps.

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Rotating windmills

1 4 ↑ 3

The following extended graphs are rotating windmills:

▶ the windmill, ▶ the four base rotating windmills, ▶ any extended graph obtained by applying one of

three construction steps.

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Rotating windmills

1 4 ↑ 3

The following extended graphs are rotating windmills:

▶ the windmill, ▶ the four base rotating windmills, ▶ any extended graph obtained by applying one of

three construction steps.

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Rotating windmills

1 4 ↑ 3

The following extended graphs are rotating windmills:

▶ the windmill, ▶ the four base rotating windmills, ▶ any extended graph obtained by applying one of

three construction steps.

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Rotating windmills

1 4 ↑ 3

The following extended graphs are rotating windmills:

▶ the windmill, ▶ the four base rotating windmills, ▶ any extended graph obtained by applying one of

three construction steps.

↑

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Rotating windmills

1 4 3

  • The following extended graphs are rotating windmills:

▶ the windmill, ▶ the four base rotating windmills, ▶ any extended graph obtained by applying one of

three construction steps.

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Rotating windmills

1 4 3

  • The following extended graphs are rotating windmills:

▶ the windmill, ▶ the four base rotating windmills, ▶ any extended graph obtained by applying one of

three construction steps.

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Rotating windmills

1 4 3

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Rotating windmills

1 4 3

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Rotating windmills

1 4 3

  • ↑
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Rotating windmills

1 4 3

  • ↑
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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋮ ⋯ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋮ ⋯ ⋮ Case 1

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋮ ⋯ ⋮ Case 1

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋮ ⋯ ⋮ Case 1

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋯ ⋮ ⋮ Case 1

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋯ ⋮ ⋮ Case 1

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋯ ⋮ ⋮ Case 1

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋯ ⋮ ⋮ Case 1

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋯ ⋮ ⋮ Case 1

✗

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋮ ⋯ ⋮ Case 2

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋮ ⋮ Case 2 ⋯

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋯ ⋮ ⋮ Case 2 ⋯

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋮ ⋯

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋯ ⋮ ⋮ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋮ ⋮ ⋯ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋮ ⋮ ⋯ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋮ ⋮ ⋯ ⋮

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SLIDE 92

Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋮ ⋮ ⋯ ⋮

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Rotating windmills

1 4 3

  • ⟹: Rotating windmills are not sliceable.

⋮ ⋯ ⋮ ⋮ ⋯ ⋮

✗

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Conclusions

Theorem

Rotating windmills are exactly the nonsliceable graphs with exactly

  • ne separating 4-cycle.
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Conclusions

Theorem

Rotating windmills are exactly the nonsliceable graphs with exactly

  • ne separating 4-cycle.

Future work:

▶ Characterize the sliceable graphs with at least two separating

4-cycles.

▶ We conjecture that the nonsliceable graphs can be

constructed by “glueing” rotating windmills together.

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Conclusions

Theorem

Rotating windmills are exactly the nonsliceable graphs with exactly

  • ne separating 4-cycle.

Future work:

▶ Characterize the sliceable graphs with at least two separating

4-cycles.

▶ We conjecture that the nonsliceable graphs can be

constructed by “glueing” rotating windmills together. Thanks!

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References I

Ko´ zmi´ nski, K. and E. Kinnen (1985). “Rectangular duals of planar graphs”. In: Networks 15.2, pp. 145–157. ISSN: 0028-3045. Mumford, E. (2008). “Drawing Graphs for Cartographic Applications”. http://repository.tue.nl/636963. PhD thesis. TU Eindhoven. Yeap, G. and M. Sarrafzadeh (1995). “Sliceable floorplanning by graph dualization”. In: SIAM J. Disc. Math. 8.2, pp. 258–280.