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Minimum Size Tree-Decompositions

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Figure 1: Given a tree G with five vertices, for any k ≥ 1, a minimum size tree-decomposition of width at most k is illustrated: s 1 (G) = 4, s 2 (G) = s 3 (G) = 2, and s k>3 (G) = 1.
Figure 2: Examples of gadgets in graph G(S, b) [9]
Figure 3: Example of the new gadget in G(S, b).
Figure 4: In a tree-decomposition (T , X ), N T (X ) = {X 1 ,· · · , X d } and for any 1 < i ≤ d, X i ∩ X ⊆ X 1
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