1. 1 Monotone Drawings of Graphs Thanks to Peter Eades www.dia.uniroma3.it/~compunet Patrizio Angelini, Enrico Colasante, Giuseppe Di Battista, Fabrizio Frati, and Maurizio Patrignani Roma Tre University, Italy

2. 2 Konstanz Univ. Bellavista Hotel direction of monotonicity

3. 3 Konstanz Univ. Petershof hotel no direction of monotonicity

4. 4 monotone paths monotone path with respect to a half-line l –each segment of the path has a positive projection onto l monotone path –there exists an l such that the path is monotone with respect to l l p1p1 p2p2

5. 5 monotone drawings of graphs monotone (straight-line) drawing of a graph G –each pair of vertices of G are joined by a monotone path monotonicity does not imply planarity l

6. 6 overview of the talk properties of monotone drawings monotone drawings of trees monotone drawings of graphs planar monotone drawings of biconnected graphs conclusions and open problems

7. 7 properties of monotone drawings each pair of adjacent edges forms a monotone path any subpath of a monotone path is monotone affine transformations preserve monotonicity each monotone path is planar –a monotone drawing of a tree is planar not monotone

8. 8 convex drawings of trees a convex drawing of a tree is such that replacing edges leading to leaves with half-lines yields a partition of the plane into convex unbounded regions [Carlson, Eppstein, GD’06]

9. 9 strictly convex drawings of trees a strictly convex drawing of a tree T is such that: –it is a convex drawing of T –each set of parallel edges of T forms a collinear path every strictly convex drawing of a tree is monotone

10. 10 slope-disjoint drawing of trees slope-disjoint drawing of a tree T –each subtree rooted at vertex v uses an interval of slopes (  v,  v ) where  v –  v <  –if u is the parent of v you have  v <  u < s(u,v) <  u <  v –if v and w are siblings (  v,  v )  (  w,  w ) =  v u v u v w

11. 11 slope-disjoint drawing of trees any slope-disjoint drawing of a tree is monotone we propose two algorithms: –BFS-based algorithm constructs a monotone drawing of a tree on a grid of area O(n 1.6 )  O(n 1.6 ) –DFS-based algorithm constructs a monotone drawing of a tree on a grid of area O(n 2 )  O(n)

12. 12 monotone drawings of graphs any graph admits a monotone drawing –consider a spanning tree T of the input graph –produce a monotone drawing of T –add the remaining edges the produced drawings may have crossings even if the input graph is planar is it possible to have planar monotone drawings of planar graphs?

13. 13 biconnected graphs a cut-vertex is a vertex such that its removal produces a disconnected graph a biconnected graph does not have cut-vertices a separation pair of a biconnected graph is a pair of vertices whose removal produces a disconnected graph a split pair is either a separation pair or a pair of adjacent vertices

14. 14 SPQR-tree u v

15. 15 SPQR-tree Q u v

16. 16 SPQR-tree Q u v

17. 17 SPQR-tree Q S u v

18. 18 SPQR-tree Q S Q u v

19. 19 SPQR-tree Q S Q P

20. 20 SPQR-tree Q Q Q Q Q S P S

21. 21 SPQR-tree Q Q Q Q Q S P Q Q Q Q Q R S

22. 22 SPQR-tree Q Q Q Q Q S P Q Q Q Q Q R S each internal node of the tree is associated with a skeleton representing its configuration the graph represented by node  into its parent is called the pertinent of  u v u v skeleton of S

23. 23 convex drawings are monotone graphs admitting strictly convex drawings [Chiba, Nishizeki, 88] –are biconnected –have an embedding such that each split pair u,v is incident to the outer face all its maximal split components, with the possible exception of edge (u,v), have at least one edge on the outer face any strictly convex drawing of a graph is monotone [Arkin, Connelly, Mitchell, SoCG ‘89] with similar techniques we show that –any non-strictly convex drawing of a graph such that each set of parallel edges forms a collinear path is monotone

24. 24 strategy for biconnected graphs we apply an inductive algorithm to the nodes of the SPQR-tree a node  of the tree is associated with a quadrilateral shape called boomerang of  and denoted by boom(  ) the pertinent of  uses a “restricted” range of slopes the boomerangs of the children of  are arranged into boom(  ) N S E W

25. 25 strategy for biconnected graphs invariants on boom(  ) –symmetric with respect to the line through W and E –angle  + 2  <  /2 N S   E W

26. 26 properties of the drawings the inductive algorithm constructs a drawing   of pert(  ) such that –   is monotone –is contained into boom(  ) with the possible exception of the edge joining the poles of  –any vertex w of pert(  ) belongs to a path that is a composition of a path toward N which is monotone with respect to d N and uses slopes in (d N - , d N +  ) a path toward S monotone w.r.t. d S and using slopes in (d S - , d S +  ) N E S W   dSdS dNdN

27. 27 base case: Q-node if  is a Q-node draw the corresponding edge as a segment from N to S N S   E W

28. 28 if  is an S-node find the intersection between the two bisectors of  angles arrange the boomerangs of the children as in the figure recur on the children using  ’ + 2  ’ <  /2 N E S W  

29. 29 if  is a P-node split boom(  ) into a suitable number of slices compute  and  for each slice recur on the children   N E S W

30. 30 if  is an R-node remove the south pole and obtain graph G’ observe that G’ admits a convex drawing into any convex polygon draw G’ into a suitable convex polygon in the upper portion of boom(  ) squeeze the drawing towards the N-E border of boom(  ) in order to make sure that the drawing uses a restricted interval of slopes compute  and  for each child recur on the children   N E S W

31. 31 admitting planar and monotone drawings admitting strictly convex drawings planar biconnected planar triconnected trees planar simply connected ? ? ? ? ? ?

32. 32 conclusions and open problems address simply connected graphs determine tight bounds on the area requirements for grid drawings of trees devise algorithms to construct monotone drawings of non-planar graphs on a grid of polynomial size construct monotone drawings of biconnected graphs in polynomial area explore strongly monotone drawings, where each pair of vertices u,v has a joining path that is monotone with respect to the line from u to v

33. 33 change my embedding if you want me monotone

34. 34 thank you thank you