Polynomial time algorithm to find an induced cycle in non-chordal graph Let $G=(V,E),n=|V|,m=|E|$ be a graph. There exists an algorithm with time complexity $O(n+m)$ to test if given graph $G$ is chordal. If I know the graph is NOT chordal, is there a algorithm to find any of the cycles without a chord of length $k\geq 4$ (running in polynomial time of course)?
Since I found a solution myself, I'm gonna post the idea. It suffices to find chordless paths. For a vertex $v$ and $u$ its neighbour, we can delete the edge $\{u,v\}$ from $G$ and find a chordless $u$-$v$ paths. Connecting these will result in a chordless cycle.
A very simple algorithm finding such paths is described here. Naive algorithm runs in time $O(n(n+m))$ per $u$-$v$ path, they further give more efficient algorithm with running time $O(n+m)$ per $u$-$v$ path.