H-Free Coloring on Graphs with Bounded Tree-Width
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. This problem is NP-complete and is expressible in monadic second order logic (MSOL). The MSOL formulation, together with Courcelle’s theorem implies linear time solvability on graphs with bounded tree-width. This approach yields an algorithm with running time \(f(||\varphi ||, t)\cdot n\), where \(||\varphi ||\) 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 \(f(||\varphi ||, t)\) on \(||\varphi ||\) can be as bad as a tower of exponentials.
In this paper, we provide an explicit combinatorial FPT algorithm to compute the \(H\)-Free Chromatic Number of a given graph G, parameterized by the tree-width of G. The techniques are also used to provide an FPT algorithm when H is forbidden as a subgraph (not necessarily induced) in the color classes of G.