J, 86 bytes
1=2#.1-((1 e.,@{@(<@(#~(-:~.)&>)@,@{@#"0;&i./@$)*/@,@{])"1 2],~$$"1,1:`[`]}"{~[:I.1-,)
The 2 long test cases time out on TIO, so I've removed them.
- Creates a list of all 0-to-1 mutated versions of the input, with the input itself as the final element in that list.
- For each of those, checks for the required minor, using brute force, returning 1 if a minor is found, 0 otherwise.
- Swaps those zeros and ones
- Convert to a binary number
- Checks if the number is 1.
- This will only be true of every 0-to-1 mutation has the minor, but the input does not.
Jonah
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