Topological Morphing of Planar Graphs Bertinoro Workshop on Graph Drawing 2012

Topological Morphing Given two planar embeddings of the same graph, how many operations do we need to morph an embedding into the other one? A planar embedding is composed of a combinatorial embedding (rotation scheme) + an external face We defined the problem as “Topological Morphing” in analogy with the “Geometric Morphing”, in which a planar drawing is morphed into another.

Geometric Morphing Geometric morphing modifies the shape of the objects in the drawing, while maintaining the topology unchanged

Topological morphing modifies the arrangement of the objects in the drawing Topological vs Geometric Morphing

State of the Art Angelini, Cortese, Di Battista, Patrignani. Topological Morphing of Planar Graphs, GD’08 Definition of two operations to modify the embedding of a biconnected graph  Flip & Skip

Operations - Flip A flip operation “flips” a subgraph with respect to a split pair

Operations - Flip A flip operation is not allowed if the subgraph to be flipped contains all the edges of the external face

Operations - Skip A skip operation moves the external face to an adjacent face with respect to a separation pair

Operations - Skip A skip operation does not change any rotation scheme

State of the Art The problem of minimizing the number of such operations is NP-Complete (Sorting by Reversals) Polynomial-time algorithms  if the combinatorial embedding is fixed  if there is no parallel component  FPT-algorithm

Open Problem Simply-connected planar graphs  Definition of allowed operations  Each configuration has to be reachable  The mental map of the user must be preserved  Study of the problem of minimizing the number of such operations Biconnected planar graphs  Give a weight to each operation depending on the size of the involved component ?