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A linear fixed parameter tractable algorithm for connected pathwidth

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Figure 1 A graph G of connected pathwidth 2 with a subgraph of connected pathwidth 3.
Figure 2 The black bullets forms the typical sequence Tseq(α) = h4, 7, 3, 9, 1, 8, 3, 6i of the sequence α = h4,6, 5, 7, 3, 5,7, 9, 4, 6, 3, 1, 4, 7,8, 5, 6, 3,4, 4, 5, 6i of black and gray diamonds.
Figure 3 The part hs i−1 , . . . s k i of a B-boundaried sequence S for some set B ⊇ {x, y, z}
Figure 4 Let T be a 2-extension of the B-boundaried sequence S. Suppose that (5, 10) is a valid insertion position with respect to some set X ⊆ B in T
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