The tree-child network inference problem for line trees and the shortest common supersequence problem for permutation strings

One strategy for inference of phylogenetic networks is to solve the phylogenetic network problem, which involves inferring phylogenetic trees first and subsequently computing the smallest phylogenetic network that displays all the trees. This approach capitalizes on exceptional tools available for inferring phylogenetic trees from biomolecular sequences. Since the vast space of phylogenetic networks poses difficulties in obtaining comprehensive sampling, the researchers switch their attention to inferring tree-child networks from multiple phylogenetic trees, where in a tree-child network each non-leaf node must have at least one child that is an indegree-one node. Three results are obtained in this work: (1) The shortest common supersequence problem remains NP-hard even for permutation strings. (2) Derived from the first result, the tree-child network inference problem is also established as NP-hard even for line trees (also known as caterpillar trees). (3) The parsimonious tree-child networks that display all the line trees are the same as those displaying all the binary trees and their hybridization number is $\Theta \left(n^3\right)$ for n taxa.