The last in-tree recognition problem asks whether a given spanning tree can be derived by connecting each vertex with its rightmost left neighbor of some search ordering. In this study, we demonstrate that the last-in-tree recognition problem for Generic Search is $\mathsf{ NP}$-complete. We utilize this finding to strengthen a complexity result from order theory. Given a partial order $\pi$ and a set of triples, the $\mathsf{ NP}$-complete intermezzo problem asks for a linear extension of $\pi$ where each first element of a triple is not between the other two. We show that this problem remains $\mathsf{ NP}$-complete even when the Hasse diagram of the partial order forms a tree of bounded height. In contrast, we give an $\mathsf{ XP}$-algorithm for the problem when parameterized by the width of the partial order. Furthermore, we show that $\unicode{ x2013}$ under the assumption of the Exponential Time Hypothesis $\unicode{ x2013}$ the running time of this algorithm is asymptotically optimal.
Categories: cs.DM cs.CC cs.DS math.CO