### Graph planarity and related topics

- Graph Drawing (Proc. GD ’99), volume 1731 of LNCS
, 1999

"... ..."

### Common-Face Embeddings of Planar Graphs

, 2001

"... Given a planar graph G and a sequence C1,...,Cq, where each Ci is a family of vertex subsets of G, we wish to find a plane embedding of G, if any exists, such that for each i ∈ {1,..., q}, there is a face Fi in the embedding whose boundary contains at least one vertex from each set in Ci. This probl ..."

Abstract
- Add to MetaCart

Given a planar graph G and a sequence C1,...,Cq, where each Ci is a family of vertex subsets of G, we wish to find a plane embedding of G, if any exists, such that for each i ∈ {1,..., q}, there is a face Fi in the embedding whose boundary contains at least one vertex from each set in Ci. This problem has applications to the recovery of topological information from geographical data and the design of constrained layouts in VLSI. Let I be the input size, i.e., the total number of vertices and edges in G and the families Ci, counting multiplicity. We show that this problem is NP-complete in general. We also show that it is solvable in O(I log I) time for the special case where for each input family Ci, each set in Ci induces a connected subgraph of the input graph G. Note that the classical problem of simply finding a planar embedding is a further special case of this case with q = 0. Therefore, the processing of the additional constraints C1,..., Cq only incurs a logarithmic factor of overhead.

### Linear-time algorithms to color topological graphs

, 2005

"... We describe a linear-time algorithm for 4-coloring planar graphs. We indeed give an O(V + E + |χ | + 1)-time algorithm to C-color V-vertex E-edge graphs embeddable on a 2-manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) ..."

Abstract
- Add to MetaCart

We describe a linear-time algorithm for 4-coloring planar graphs. We indeed give an O(V + E + |χ | + 1)-time algorithm to C-color V-vertex E-edge graphs embeddable on a 2-manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) time, to find the exact chromatic number of a maximal planar graph (one with E = 3V − 6) and a coloring achieving it. Finally, there is a linear-time algorithm to 5-color a graph embedded on any fixed surface M except that an M-dependent constant number of vertices are left uncolored. All the algorithms are simple and practical and run on a deterministic pointer machine, except for planar graph 4-coloring which involves enormous constant factors and requires an integer RAM with a random number generator. All of the algorithms mentioned so far are in the ultra-parallelizable deterministic computational complexity class “NC. ” We also have more practical planar 4-coloring algorithms that can run on pointer machines in O(V log V) randomized time and O(V) space, and a very simple deterministic O(V)-time coloring algorithm for planar graphs which conjecturally uses 4 colors.

### A Faster Algorithm for Torus Embedding

, 2006

"... Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and di ..."

Abstract
- Add to MetaCart

Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the well-studied problem of finding the complete set of torus obstructions.

### Subgraph Homeomorphism via the Edge Addition Planarity Algorithm

, 2012

"... This paper extends the edge addition planarity algorithm from Boyer and Myrvold to provide a new way of solving the subgraph homeomorphism problem for K2,3, K4, and K3,3. These extensions derive much of their behavior and correctness from the edge addition planarity algorithm, providing an alternati ..."

Abstract
- Add to MetaCart

This paper extends the edge addition planarity algorithm from Boyer and Myrvold to provide a new way of solving the subgraph homeomorphism problem for K2,3, K4, and K3,3. These extensions derive much of their behavior and correctness from the edge addition planarity algorithm, providing an alternative perspective on these subgraph homeomorphism problems based on affinity with planarity rather than triconnectivity. Reference implementations of these algorithms have been made available in an open source project

### New 3D Fourier Descriptors for Genus-zero Mesh Objects

"... Abstract. The 2D Fourier Descriptor is an elegant and powerful technique for 2D shape analysis. This paper intends to extend such technique to 3D. Though conceptually natural, such an extension is not trivial in that two critical problems, the spherical parametrization and invariants construction, m ..."

Abstract
- Add to MetaCart

Abstract. The 2D Fourier Descriptor is an elegant and powerful technique for 2D shape analysis. This paper intends to extend such technique to 3D. Though conceptually natural, such an extension is not trivial in that two critical problems, the spherical parametrization and invariants construction, must be solved. By using a newly developed surface parametrization method–the discrete conformal mapping (DCM)—we propose a 3D Fourier Descriptor (3D-FD) for representing and recognizing arbitrarily-complex genus-zero mesh objects. A new DCM algorithm is suggested which solves the first problem efficiently. We also derive a method to construct a truly complete set of Spherical Harmonic invariants. The 3D-FD descriptors have been tested on different complex mesh objects. Experiment results for shape representation are satisfactory. 1

### Optimizing and Visualizing Planar Graphs via Rectangular Dualization

"... Graphs are among the most studied and used mathematical tools for representing data and their relationships. The increasing number of applications exploiting their descriptive power makes graph visualization a key issue in modern graph theory and computer science. This is witnessed by the widespread ..."

Abstract
- Add to MetaCart

Graphs are among the most studied and used mathematical tools for representing data and their relationships. The increasing number of applications exploiting their descriptive power makes graph visualization a key issue in modern graph theory and computer science. This is witnessed by the widespread interest Graph Drawing has been able to excite in the research community over the past decade. Ever more often software offers a valuable support to key fields of science (such as chemistry, biology, physics and mathematics) in elaborating and interpreting huge quantities of data. Usually, however, a drawing explains the nature of a phenomenon better than hundreds of tables and sterile figures; that is why visualization is so important in modern Computer Science. Graph Drawing is an important part of Information Visualization addressing specific visualization problems on graphs. Most of the research work in Graph Drawing aims at creating representations complying with aesthetic criteria, relying on the principle that a drawing nice and pleasant to see is also effective. Difficulties arise when the size of graphs blows up; in this case appropriate optimization techniques must be used before visualization. One widely used optimization

### International Journal, Pattern Recognition Conformal Spherical Representation of 3D Genus-Zero Meshes

"... This paper describes an approach of representing 3D shape by using a set of invariant Spherical Harmonic (SH) coefficients after conformal mapping. Specifically, a genus-zero 3D mesh object is first conformally mapped onto the unit sphere by using a modified discrete conformal mapping, where the mod ..."

Abstract
- Add to MetaCart

This paper describes an approach of representing 3D shape by using a set of invariant Spherical Harmonic (SH) coefficients after conformal mapping. Specifically, a genus-zero 3D mesh object is first conformally mapped onto the unit sphere by using a modified discrete conformal mapping, where the modification is based on Möbius Factorization and is aimed at obtaining a canonical conformal mapping. Then a Spherical Harmonic Analysis is applied to the resulting conformal spherical meshes. The obtained SH coefficients are further made invariant to translation and rotation, while at the same time retain their completeness, so that the original shape information has been faithfully preserved.