Abstract
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 $C_u$ and $C_v$. 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 $\cO(2^{\mathcal{O}(h)} \cdot n + n^{2}\cdot \log n)$ a model of $H$ in $G$, or correctly concludes that $G$ does not contain $H$ as a minor. Our algorithm is the first {\sl 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 \emph{partially embedded dynamic programming