Distinguished minterm

Also called: distinguished 1-cell, distinguished cell, distinguished 1, distinguished one

A 1 on a K-map that is covered by exactly one prime implicant. Any prime implicant that covers a distinguished minterm is essential.

A distinguished minterm (or distinguished 1-cell) is a 1 that lies inside exactly one prime implicant. No other maximal group reaches it.

This is the test that decides which groups are essential: a prime implicant is an essential prime implicant exactly when it covers at least one distinguished minterm. Since that 1 has to be covered, and only one group can do it, that group must appear in every minimal solution.

How to find them:

List every prime implicant.
For each 1, count how many prime implicants contain it.
The 1s with a count of exactly one are distinguished.

Two details:

Only real 1s count. A don't-care never needs covering, so it can't be distinguished.
A 1 can sit in several groups and still not be distinguished, even if it looks central. Being covered by many groups is the opposite of distinguished.

When no 1 is distinguished, there are no essential prime implicants, and the map is cyclic.

A\BC	00	01	11	10
0	1m0	0m1	1m3	1m2
1	0m4	0m5	0m7	1m6

Worked example

Example

Finding distinguished minterms

F = Σm(0, 2, 3, 6), shown above.

1.
Prime implicants: (m0 m2), (m2 m3), (m2 m6). No group of 4 fits.
2.
m0 is only in : distinguished.
3.
m3 is only in : distinguished.
4.
m6 is only in : distinguished.
5.
m2 is in all three, so it is not distinguished.
6.
All three prime implicants are essential: F = .

Common mistakes

Counting a don't-care as distinguished. Only cells that must be 1 count.
Counting non-prime groups when checking coverage. Only maximal groups are compared.
Assuming the 1 in the middle of many groups is the important one. It's the 1 with only one group that matters.