In an undirected graph, a matching cut is a partition of vertices into two sets such that the edges across the sets induce a matching. The Matching Cut problem is the problem of deciding whether a given graph has a matching cut.
Let $H$ be a fixed undirected graph. A vertex coloring of an undirected input graph $G$ is said to be an $H$-Free Coloring if none of the color classes contain $H$ as an induced subgraph. The $H$-Free Chromatic Number of $G$ is the minimum number of colors required for an $H$-Free Coloring of $G$.
Both the Matching Cut problem and $H$-Free Coloring problem can be expressed using a monadic second order logic (MSOL) formula and hence is solvable in linear time for graphs with bounded tree-width. However, this approach leads to a running time $f(\lVert \varphi \rVert, t) n^{O(1)}$, where $\lVert \varphi \rVert$ is the length of the MSOL formula, $t$ is the tree-width of the graph and $n$ is the number of vertices of the graph. The dependency of $f(\lVert \varphi \rVert, t)$ on $\lVert \varphi \rVert$ can be as bad as a tower of exponentials.
In this paper, we provide explicit combinatorial FPT algorithms for Matching Cut problem and$H$-Free Coloring problem, parameterized by the tree-width of $G$. The single exponential FPT algorithm for the Matching Cut problem answers an open question posed by Kratsch and Le (2016). The techniques used in the paper are also used to provide an FPT algorithm for a variant of $H$-Free Coloring, where $H$ is forbidden as a subgraph (not necessarily induced) in the color classes of $G$.