# Domino treewidth

## Abstract

We consider a special variant of tree-decompositions, called *domino tree-decompositions*, 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 *C*_{ k,d } such that a graph with treewidth at most *k* and maximum degree at most *d* has domino treewidth at most *C*_{ k,d } The domino treewidth of a tree can be computed in *O*(*n*^{2} log *n*) time. There exist polynomial time algorithms that — for fixed *k* — decide whether a given graph *G* has domino treewidth at most *k*. If *k* is not fixed, this problem is NP-complete. The domino treewidth problem is hard for the complexity classes *W*[*t*] for all *t* ξ **N**, and hence the problem for fixed *k* is unlikely to be solvable in *O*(*n*^{ c }), where *c* is a constant, not depending on *k*.

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