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

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## 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.

## Keywords

Polynomial Time Small Tree Linear Chain Vertex Cover Polynomial Time Approximation Scheme## Preview

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