An FPT algorithm for the embeddability of graphs into two-dimensional simplicial complexes

@inproceedings{Verdire2021AnFA,
  title={An FPT algorithm for the embeddability of graphs into two-dimensional simplicial complexes},
  author={{\'E}ric Colin de Verdi{\`e}re and Thomas Magnard},
  booktitle={ESA},
  year={2021}
}
We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f(c)n, where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of… Expand

References

SHOWING 1-10 OF 31 REFERENCES
Embedding graphs into two-dimensional simplicial complexes
TLDR
This work considers the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C, and reduces the problem to an embedding extension problem on a surface. Expand
A Simpler Linear Time Algorithm for Embedding Graphs into an Arbitrary Surface and the Genus of Graphs of Bounded Tree-Width
TLDR
This paper gives a new linear time algorithm which computes the genus and constructs minimum genus embeddings of graphs of bounded tree-width and resolves a conjecture by Neil Robertson and solves one of the most annoying long standing open question about complexity of algorithms onGraph minors theory. Expand
Deleting Vertices to Graphs of Bounded Genus
We show that a problem of deleting a minimum number of vertices from a graph to obtain a graph embeddable on a surface of a given Euler genus is solvable in time $$2^{C_g \cdot k^2 \log k}Expand
On the Homotopy Test on Surfaces
  • F. Lazarus, J. Rivaud
  • Computer Science, Mathematics
  • 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
  • 2012
TLDR
This work presents a geometric approach, based on previous works by Colin de Verdière and Erickson, that provides optimal homotopy tests and describes linear time algorithms to decide if c and d are homotopic in S, either freely or with fixed base point. Expand
Computing crossing number in linear time
We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has crossing number at most k, and if this is the case, computes a drawing of the graph inExpand
Fully Polynomial-Time Parameterized Computations for Graphs and Matrices of Low Treewidth
TLDR
An approximation algorithm for treewidth with time complexity suited to the running times of polynomial time is given, which shows that the existence of algorithms with similar running times is unlikely for the problems of finding the diameter and the radius of a graph of low Treewidth. Expand
A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface
  • B. Mohar
  • Mathematics, Computer Science
  • SIAM J. Discret. Math.
  • 1999
TLDR
A linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homeomorphic to a minimal forbidden subgraph for embeddability in S that yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbiddenSubgraphs. Expand
An additivity theorem for the genus of a graph
  • G. Miller
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. B
  • 1987
TLDR
A new combinatorial representation of an embedding which seems to lend itself to combinatorsial arguments for embedding of graphs on unoriented surfaces is introduced and it is shown that (generalized) genus is additive over edge (vertex) amalgams. Expand
Dynamic generators of topologically embedded graphs
TLDR
This work provides a data structure for maintaining an embedding of a graph on a surface and computing generators of the fundamental group of the surface, in amortized time O(log n + log g(log log g)3) per update on asurface of genus g, and applies similar ideas to improve the constant factor in a separator theorem for low-genus graphs, and to find in linear time a tree-decomposition of low-generation low-diameter graphs. Expand
Polynomial bounds for the grid-minor theorem
TLDR
The first polynomial relationship between treewidth and grid-minor size is obtained by showing that f(k) = Ω(kδ) for some fixed constant δ > 0, and an algorithm is described that finds a model of such a grid-Minor in G. Expand
...
1
2
3
4
...