What is the smallest 3-regular graph to have a unique perfect matching?
With a large enough number of nodes, it is possible for a 3-regular graph to have no perfect matching (example can be seen in this question Cubic graphs without a perfect matching and a vertex incident to three bridges ). So I believe 3-regular graphs with a unique matching likely exists, but I am unsure how to go about constructing and proving what the smallest one is. Likely there is no better answer than to brute force check all the possibilities, so I am hoping someone happens to know what this graph looks like.
Even better: Does anyone know of an online searchable graph database that allows searching for small graphs with certain properties?