  A framework consists of an undirected graph $G$ and a matroid $M$ whose
elements correspond to the vertices of $G$. Recently, Fomin et al. [SODA 2023]
and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing
paths of rank $k$ in frameworks. More precisely, for vertices $s$ and $t$ of
$G$, and an integer $k$, they gave FPT algorithms parameterized by $k$ deciding
whether there is an $(s,t)$-path in $G$ whose vertex set contains a subset of
elements of $M$ of rank $k$. These algorithms are based on Schwartz-Zippel
lemma for polynomial identity testing and thus are randomized, and therefore
the existence of a deterministic FPT algorithm for this problem remains open.
We present the first deterministic FPT algorithm that solves the problem in
frameworks whose underlying graph $G$ is planar. While the running time of our
algorithm is worse than the running times of the recent randomized algorithms,
our algorithm works on more general classes of matroids. In particular, this is
the first FPT algorithm for the case when matroid $M$ is represented over
rationals. Our main technical contribution is the nontrivial adaptation of the
classic irrelevant vertex technique to frameworks to reduce the given instance
to one of bounded treewidth. This allows us to employ the toolbox of
representative sets to design a dynamic programming procedure solving the
problem efficiently on instances of bounded treewidth.
