Monotone Drawings of Graphs

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

direction of monotonicity Konstanz Univ. Bellavista Hotel

no direction of monotonicity Konstanz Univ. Petershof hotel

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 p2 p1 l

l 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

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

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

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]

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

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) =  u u v w v v

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(n1.6)  O(n1.6) • DFS-based algorithm • constructs a monotone drawing of a tree on a grid of area O(n2)  O(n)

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?

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

SPQR-tree v u

SPQR-tree v Q u

SPQR-tree v Q u S

SPQR-tree v Q u S Q

SPQR-tree Q S P Q

SPQR-tree Q S P Q S Q Q Q

SPQR-tree Q S P Q R Q S Q Q Q Q Q Q Q

v u skeleton of S SPQR-tree v • 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  Q u S P Q R Q S Q Q Q Q Q Q Q

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

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 E W S

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

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 dN and uses slopes in (dN-, dN+) • a path toward S monotone w.r.t. dS and using slopes in (dS-, dS+) N dN   E W dS S

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

if  is an S-node N  • 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  E W S

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

if  is an R-node N • 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   E W S

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

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

change my embedding if you want me monotone

thank you

Monotone Drawings of Graphs