Abstract We consider the problem of inferring an edge-labeled graph from the sequence of edge labels seen in a walk on that graph. It has been known that this problem is solvable in $O( n \log{n})$ time when the targets are path or cycle graphs. This paper presents an online algorithm for the problem of this restricted case that runs in $O ( n )$ time, based on Manacher’s algorithm for computing all the maximal palindromes in a string.