Fast Minor Testing in Planar Graphs

@article{Adler2011FastMT,
  title={Fast Minor Testing in Planar Graphs},
  author={Isolde Adler and F. Dorn and F. Fomin and Ignasi Sau and D. Thilikos},
  journal={Algorithmica},
  year={2011},
  volume={64},
  pages={69-84}
}
Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G between components Cu and Cv. A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time… Expand
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References

SHOWING 1-10 OF 48 REFERENCES
Constant-Factor Approximations of Branch-Decomposition and Largest Grid Minor of Planar Graphs in O(n1 + ε) Time
TLDR
Constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs and an algorithm which constructs in O(n^{1+\frac{1}{c}}\log n) time a branch-Decomposition of G with width at most $\alpha\,{\mathop{\rm bw}}(G)$. Expand
Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in O(n1+ϵ) time
TLDR
The constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs and an algorithm which constructs a gxg grid minor of G with g>=gm(G)@b in O(n^1+^1^clogn) time are given. Expand
Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size
TLDR
It is shown that for any constant c<2, the bound of  bw(G) does not hold in general for a planar graph G, and implies quadratic time constant-factor approximation algorithms for planar graphs for both problems of finding a largest grid minor and of finding an optimal branch-decomposition. Expand
Faster parameterized algorithms for minor containment
TLDR
This work improves the dependence on k of Hicks' result by showing that checking if H is a minor of G can be done in time, and obtains the first single-exponential algorithm for minor containment testing. Expand
Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions
TLDR
The approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions to design fast subexponential exact and parameterized algorithms onPlanar graphs. Expand
Linear time algorithm for isomorphism of planar graphs (Preliminary Report)
TLDR
The time bound for planar graph isomorphism is improved to O(|V|) time and the algorithm can be easily extended to partition a set of planar graphs into equivalence classes of isomorphic graphs in time linear in the total number of vertices in all graphs in the set. Expand
Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs
TLDR
An algorithm is presented that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) where c=4^ 6\sqrt 34 and k is the size of the face cover set. Expand
Complexity of Disjoint Paths Problems in Planar Graphs
TLDR
This work considers some complexity results for problems in planar graphs for fixed k and proves that the edge-disjoint undirected problem can be solved in linear time. Expand
Approximation algorithms for NP-complete problems on planar graphs
  • B. Baker
  • Mathematics, Computer Science
  • 24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
  • 1983
TLDR
A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. Expand
Planar Subgraph Isomorphism Revisited
  • F. Dorn
  • Mathematics, Computer Science
  • STACS
  • 2010
TLDR
Eppstein gives the first linear time algorithm for subgraph isomorphism for a fixed-size pattern, say of order $k$, and arbitrary planar host graph, and introduces the technique of ``embedded dynamic programming'' on a suitably structured graph decomposition. Expand
...
1
2
3
4
5
...