Fast Minor Testing in Planar Graphs

Isolde Adler (1) , Frederic Dorn (1) , Fedor V. Fomin (1) , Ignasi Sau (2) , Dimitrios M. Thilikos (3)

1 UiB - Department of Informatics [Bergen]
2 ALGCO - Algorithmes, Graphes et Combinatoire
3 DI NKUA - Department of Informatics and Telecomunications [Kapodistrian Univ]

Minor containment is a fundamental problem in Algorithmic Graph Theory, as numerous graph algorithms use it as a subroutine. 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. 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 2O(h)ċn+O(n2ċlog n) a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first single-exponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming.

Mots clés: Graph minors; Planar graphs; Branchwidth; Parameterized complexity; Dynamic programming

Domaines: Mathématique discrète [cs.DM]

Isolde Adler, Frederic Dorn, Fedor V. Fomin, Ignasi Sau, Dimitrios M. Thilikos. Fast Minor Testing in Planar Graphs. ESA 2010 - 18th Annual European Symposium on Algorithms, Sep 2010, Liverpool, United Kingdom. pp.97-109, ⟨10.1007/978-3-642-15775-2_9⟩. ⟨lirmm-00736769⟩