Tree decomposition
Tree with 'bags' of graph vertices at each node. Every edge in some bag. For each vertex, bags containing it form connected subtree.
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Width
Max bag size minus 1. Minimize over decompositions.
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Courcelle's theorem
Any MSO-definable property on bounded-treewidth graphs solvable in linear time. Huge meta-theorem.
Computation
Treewidth exact: NP-hard. Approximation: O(log OPT) polynomial. FPT algorithms in treewidth.