Graph inference from a walk for trees of bounded degree 3 is NP-complete

Abstract

The graph inference from a walk for a class C of undirected edge-colored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ⩾ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(n log n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.

This work is partly supported by Grant-in-Aid for Scientific Research on Priority Areas “Genome Informatics” from the Ministry of Education, Science and Culture, Japan.

This author is a Research Fellow of the Japan Society for the Promotion of Science (JSPS). The author's research is partly supported by Grants-in-Aid for JSPS research fellows from the Ministry of Education, Science and Culture, Japan.