A well known conjecture asks about the existence of dominating circuits in cubic graphs, where a dominating circuit $C$ in a cubic graph $G$ is a circuit such that all edges in G have at least one endvertex in $C$. I am wondering about a relaxed version of this. The question I have is: given a cubic, 3-connected graph $G$, what can we say about the existence of a circuit $C$ in $G$ such that $G-V(C)$ is a single, not necessarily induced path. Are there any results in this direction?
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