Finding branch-decomposition and rank-decomposition
Petr Hliněný, Sang‐il Oum · SIAM Journal on Computing · 2008
We present a new algorithm that can output the rank-decomposition of width at most $k$ of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most $k$ if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixed-parameter tractable, that is, they run in time $O(n^3)$ for each fixed value of $k$ where $n$ is the number of vertices / elements of the input.