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Carving-width, tree-width and area-optimal planar graph drawing Therese Biedl University of Waterloo biedl@uwaterloo.ca May 5, 2014 Therese Biedl Carving-width, tree-width and graph drawing 1 / 19 Graph Drawing Given: A graph G = ( V , E

  1. Carving-width, tree-width and area-optimal planar graph drawing Therese Biedl University of Waterloo biedl@uwaterloo.ca May 5, 2014 Therese Biedl Carving-width, tree-width and graph drawing 1 / 19

  2. Graph Drawing Given: A graph G = ( V , E ). Therese Biedl Carving-width, tree-width and graph drawing 2 / 19

  3. Graph Drawing Given: A graph G = ( V , E ). Want: a pretty drawing (created by an algorithm.) Therese Biedl Carving-width, tree-width and graph drawing 2 / 19

  4. Graph Drawing Given: A graph G = ( V , E ). Want: a pretty drawing (created by an algorithm.) What is “pretty”? Therese Biedl Carving-width, tree-width and graph drawing 2 / 19

  5. Graph Drawing Given: A graph G = ( V , E ). Want: a pretty drawing (created by an algorithm.) What is “pretty”? This talk: Straight-line drawing, planar graph, no crossing. Therese Biedl Carving-width, tree-width and graph drawing 2 / 19

  6. Previous work Every planar graph has a planar straight-line drawing. (Wagner’36, F´ ary’48, Stein’50 ) Therese Biedl Carving-width, tree-width and graph drawing 3 / 19

  7. Previous work Every planar graph has a planar 8 rows = straight-line drawing. height 8 (Wagner’36, F´ ary’48, Stein’50 ) Want small integer coordinates. Therese Biedl Carving-width, tree-width and graph drawing 3 / 19

  8. Previous work Every planar graph has a planar 8 rows = straight-line drawing. height 8 (Wagner’36, F´ ary’48, Stein’50 ) Want small integer coordinates. Theorem (de Fraysseix, Pach, Pollack 1990; Schnyder 1990) Every planar graph can be drawn in an O ( n ) × O ( n ) -grid. Therese Biedl Carving-width, tree-width and graph drawing 3 / 19

  9. Previous work Theorem (de Fraysseix, Pach, Pollack 1990; Schnyder 1990) Every planar graph can be drawn in an O ( n ) × O ( n ) -grid. Theorem (de Fraysseix, Pach, Pollack 1988) Some planar graphs need an Ω( n ) × Ω( n ) -grid. Therese Biedl Carving-width, tree-width and graph drawing 4 / 19

  10. Previous work Theorem (de Fraysseix, Pach, Pollack 1990; Schnyder 1990) Every planar graph can be drawn in an O ( n ) × O ( n ) -grid. Theorem (de Fraysseix, Pach, Pollack 1988) Some planar graphs need an Ω( n ) × Ω( n ) -grid. Many planar graph drawing results since: Drawing in an n × n -grid (Schnyder 1990) Drawing in a 2 3 n × O ( n )-grid (Chrobak, Nakano 1994) 9 n 2 (Brandenburg 2008) Drawing in area 8 4 n 2 (Miura et al. 1999) 4-connected planar graphs in area 1 Trees in O ( n log n ) area (easy). Outer-planar graphs in O ( n log n ) area (B. 2002) Series-parallel graphs in O ( n 1 . 5 ) area (B. 2009) and many more like that.... Therese Biedl Carving-width, tree-width and graph drawing 4 / 19

  11. Area-optimal planar graph drawing? Most graph drawing results have the form: Algorithm draws graph in class X with area f ( n ) . Some graph in class X needs area Ω( f ( n )) . Therese Biedl Carving-width, tree-width and graph drawing 5 / 19

  12. Area-optimal planar graph drawing? Most graph drawing results have the form: Algorithm draws graph in class X with area f ( n ) . Some graph in class X needs area Ω( f ( n )) . Very few graph drawing results of the form: Algorithm draws G with optimal area for G. (Or at least within constant factor.) Therese Biedl Carving-width, tree-width and graph drawing 5 / 19

  13. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  14. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  15. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...) Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  16. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...) Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  17. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...) Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  18. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...) Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  19. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...) Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  20. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...) Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  21. Area-optimal planar graph drawing? Problem DrawOptArea : Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...) Why are Apollonian networks easy? treewidth 3? faces are triangles? both? Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

  22. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  23. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  24. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  25. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  26. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  27. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  28. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  29. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  30. Treewidth Definition Treewidth tw ( G ) = min { k : G has chordal super-graph with clique-size k } − 1 G has bounded treewidth ⇒ many NP-hard problems become polynomial (FPT). Often a first step towards developing a PTAS. Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

  31. Planar graph drawing and treewidth DrawOptArea is NP-hard. Polynomial if treewidth bounded? Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

  32. Planar graph drawing and treewidth DrawOptArea is NP-hard. Polynomial if treewidth bounded? No! (Reduction-graph in [KW07] for NP-hardness of DrawOptArea .) Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

  33. Planar graph drawing and treewidth DrawOptArea is NP-hard. Polynomial if treewidth bounded? No! (Reduction-graph in [KW07] for NP-hardness of DrawOptArea .) Graph is drawn on 3 rows. Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

  34. Planar graph drawing and treewidth DrawOptArea is NP-hard. Polynomial if treewidth bounded? No! (Reduction-graph in [KW07] for NP-hardness of DrawOptArea .) Graph is drawn on 3 rows. Therefore [FLW03]: pathwidth ≤ 3. Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

  35. Planar graph drawing and treewidth DrawOptArea is NP-hard. Polynomial if treewidth bounded? No! (Reduction-graph in [KW07] for NP-hardness of DrawOptArea .) Graph is drawn on 3 rows. Therefore [FLW03]: pathwidth ≤ 3. Well-known: then treewidth ≤ 3. Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

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