Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista,

Fabrizio Frati, Maurizio Patrignani, Vincenzo Roselli

Graph G = (V,E)

Vertices are points

Edges are straight-line segments

Planar: edges do not cross each other

Metamorphosis I -- M. C. Escher

Morphing is a special effect in motion pictures and animations that changes (or morphs) one image or shape into another through a seamless transition.

(Wikipedia)

… changes (or morphs) one planar graph drawing into another through a seamless transition

Vertices move at uniform speed along straight-line segments

Planarity is preserved during the whole morph

(…)

Some pairs of drawings require several steps

The complexity of a morphing algorithm is defined as the number of steps it performs

Tietze, Rend. Circ. Matematico di Palermo, 38(1):247-304, 1914

We are in the centennial of the Morphing problem!

Smith, 1917

Veblen, 1917

Alexander, 1923

Cairns, 1944

A planar morph exists between any two planar drawings of any maximal planar graph(with an exponential number of steps)

Thomassen, Journal of Combinatorial Theory B, 1983

A planar convex morph exists between any two planar convex drawings of any maximal planar graph,

(with an exponential number of steps)

Aicholzer et al, 2011

Floater, Gotsman, Surazhskyseveral papers

Lubiw Petrick, 2011

Biedl et al., 2006

Alamdari et al., SODA 2013

A planar morph exists between any two planar drawings of any maximal planar graph,

(with a polynomial numer of steps)

O(n2) steps for maximal planar

O(n4) steps for maximal planar

Angelini et al., GD 2013

Barrera-Cruz et al., MCDMCG, 2013

Angelini et al., GD 2013

Barrera-Cruz et al., MCDMCG, 2013

At each morphing step, all the vertices move along the same direction

Angelini et al., GD 2013

Barrera-Cruz et al., MCDMCG, 2013

At each morphing step, all the vertices move along the same direction

Maximal Planar graphs in O(n2) stepsSeries-parallel graphs in O(n) steps

Theorem 1: There exist an algorithm to morph planar graph drawings in O(n) steps

Theorem 2: There exist two drawings of a planar graph (a path) such that morphing one into the other requires at least Ω(n) steps

Every planar graph contains at least a vertex of degree at most 5 with no chord between its neighbors

By Euler’s formula, |E| <= 3n-6

Every polygon with at most 5 vertices has at least one vertex on the boundary of its kernel

Every polygon with at most 5 vertices has at least one vertex on the boundary of its kernel

Contraction of a low-degree vertex on one of its neighbors

1: Select a vertex v of low degree.

It might be contractibleon different neighborsin the two drawings

vv

2: Contract v in boththe drawings.

We obtain differentgraphs!

vv

O(1) O(1)

3: Remove v and compute a drawing in which the polygon of itsneighbors is convex

vv

O(1) O(1)

In this drawing, passing from one neighbor to the other is always possible!

vv

O(1) O(1)

In this drawing, passing from one neighbor to the other is always possible!

vv

O(1) O(1)

In this drawing, passing from one neighbor to the other is always possible!

vv

O(1) O(1)

O(1)

Recursively morph the two contracted drawings to the convex ones in T(n-1) steps

T(n) = 2 T(n-1) + O(1)

= O(2n)

vv

T(n-1) T(n-1)

O(1) O(1)

O(1)

The morph is not to a specific convex drawing!v

v

O(n) T(n-1)

O(1) O(1)

O(1)

The morph is not to a specific convex drawing!

T(n) = T(n-1) + O(n)

= O(n2)

vv

T(n-1)

O(1) O(1)

O(n)O(1)

The morph is to a (very) specific convex drawing!v

v

O(1) T(n-1)

O(1) O(1)

O(1)

The morph is to a (very) specific convex drawing!

T(n) = T(n-1) + O(1)

= O(n)

vv

T(n-1)

O(1) O(1)

O(1)O(1)

Given:

a triconnected planar graph G = (V,E)

A set L of parallel lines

A mapping of the vertices of V to lines of L such that orienting the edges of E according to the order of the lines in L yields an st-orientation of G

st-orientation: each face is composed of two oriented paths

Then, G admits a convex drawing in which each vertex of V lies on the line of L it is mapped to

Hong, Nagamochi. J. of Discrete Algorithms, 2010

Hong, Nagamochi. J. of Discrete Algorithms, 2010

A linear number of steps are necessary to morph one of these drawings into the other

There exists an edge that must perform a linear number of complete rotations

In a single step, only (a bit less than) half rotation can be performed

All the algorithms produce intermediate drawings on a very large grid

What about morphing on a limited-size grid?

Can we avoid contractions?

Can we produce convexity-preserving morphings with a polynomial number of steps?

How to morph non-planar graphs?