# A lower bound for treewidth and its consequences

## Abstract

We present a new lower bound for the treewidth (and hence the pathwidth) of a graph and give a linear-time algorithm to compute the bound. With the growing interest in treewidth based methods, this bound has many potential applications.

Our bound helps shed new light on the structure of obstructions for width *ω*. As a result, we are able to characterize completely those treewidth obstructions of order *ω*+3. Unexpectedly, we find that these graphs are exactly the pathwidth obstructions of order *ω*+3. Further, we are also able to enumerate these obstructions.

Surprisingly, while there is only one obstruction of order *ω*+2 for width *ω*, we find that the number of obstructions of order *ω*+3 alone is an asymptotically exponential function of *ω*. Our proof of this is based on the theory of partitions of integers and is the first non-trivial lower bound on the number of obstructions for treewidth.

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