# Odd holes and odd walks

Given a graph and two vertices can the following problem be solved in polynomial time ?

Find a triangle-free odd -walk in , where a walk can also contain repetitions of edges, and triangle-free means that the vertex-set of the walk does not contain any triangle (but can contain an odd hole).

A polynomial algorithm for this problem would specialize to a polynomial algorithm for finding odd holes (and even pairs in odd-hole-free graphs). Bienstock proved that it is NP-hard to find odd holes containing a given . However, a triangle-free odd -walk exists in a -connected graph if and only if there exists an odd hole in (not necessarily containing ).

Contributed by András Sebo and Nicolas Trotignon

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