Domino Treewidth

We consider a special variant of tree-decompositions, called domino treedecompositions, and the related notion of domino treewidth. In a domino tree-decomposition, each vertex of the graph belongs to at most two nodes of the tree. We prove that for every k, d, there exists a constant ck;d such that a graph with treewidth at most k and maximum degree at most d has domino treewidth at most ck;d. The domino treewidth of a tree can be computed in O(n logn) time. There exist polynomial time algorithms that | for xed k | decide whether a given graphG has domino treewidth at most k. If k is not xed, this problem is NP-complete. The domino treewidth problem is hard for the complexity classes W [t] for all t 2 N, and hence the problem for xed k is unlikely to be solvable in O(n), where c is a constant, not depending on k.