Kernelization Rules for Special Treewidth and Spaghetti Treewidth

Using the framework of kernelization we study whether efficient preprocessing schemes for the Special Treewidth problem can give provable bounds on the size of the processed instances. In this paper it is shown that Special Treewidth has a kernel with O(`) vertices, where ` denotes the size of a vertex cover. This implies that given an instance (G, k) of Special Treewidth we can efficiently reduce its size to O((`∗)3) vertices, where ` is the size of a vertex cover in G. Next we provide a characterization of the special-partial 2-trees, the class of graphs bounded by Special Treewidth 2, using the notion of mambas and Paths of Cycles. We also introduce a new cousin of Treewidth and Special Treewidth, the Spaghetti Treewidth problem. It is shown that Spaghetti Treewidth also has a kernel with O(`) vertices, where l denotes the size of a vertex cover.