In this talk, I will present a framework combining the rank-based approach with the notion of $d$-neighbor equivalence. The rank-based approach is a technique introduced by Bodlaender et al. in 2015 to obtained $2^{O(k)}\cdot n$ time algorithms, $k$ the treewidth of the input graph, for a wide range of connectivity problems.

The $d$-neighbor equivalence is a tools introduced by Bui-Xuan et al. in 2013 to obtained efficient parameterized algorithms for many width measures (clique-width, rank-width, mim-width,…) and for many problems with a locally checkable constraint (Dominating Set, Independent Set,…).

By combining these two notions, we obtain efficient algorithms for several connectivity problems such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$ and $n^{O(k)}$ time algorithms parameterized respectively by clique-width, $\mathbb{Q}$-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the best time complexity for Vertex Cover and Dominating Set.

Paper available on HAL : https://hal.archives-ouvertes.fr/hal-01799573v2/document