The K7 family consists of 20 graphs (diagram at bottom). Six of them, Cousins 9, 14, 16, 17, 19, and 20, are not IK. The remaining 14 are MMIK.
K_7 is Cousin 1.
Cousin 1:
{{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{2,3},{2,4},{2,5},{2,6},{2,7},{3,4},{3,5},{3,6},{3,7},{4,5},{4,6},{4,7},{5,6},{5,7},{6,7}}
Cousin 2:
{{1,4},{1,5},{1,6},{1,7},{1,8},{2,4},{2,5},{2,6},{2,7},{2,8},{3,4},{3,5},{3,6},{3,7},{3,8},{4,5},{4,6},{4,7},{5,6},{5,7},{6,7}}
Cousin 3:
{{1,6},{1,7},{1,8},{1,9},{2,4},{2,5},{2,6},{2,7},{2,8},{3,4},{3,5},{3,6},{3,7},{3,8},{4,6},{4,7},{4,9},{5,6},{5,7},{5,9},{6,7}}
Cousin 4:
{{1,4},{1,5},{1,6},{1,7},{1,8},{2,4},{2,5},{2,6},{2,7},{2,8},{3,4},{3,5},{3,6},{3,7},{3,8},{4,7},{4,9},{5,7},{5,9},{6,7},{6,9}}
Cousin 5:
{{1,8},{1,9},{1,10},{2,4},{2,5},{2,6},{2,7},{2,8},{3,4},{3,5},{3,6},{3,7},{3,8},{4,6},{4,7},{4,9},{5,6},{5,7},{5,9},{6,10},{7,10}}
Cousin 6:
{{1,6},{1,7},{1,8},{1,9},{2,5},{2,7},{2,8},{2,10},{3,4},{3,5},{3,6},{3,7},{3,8},{4,7},{4,9},{4,10},{5,6},{5,7},{5,9},{6,7},{6,10}}
Cousin 7:
{{1,6},{1,7},{1,8},{1,9},{2,4},{2,5},{2,8},{2,10},{3,4},{3,5},{3,6},{3,7},{3,8},{4,6},{4,7},{4,9},{5,6},{5,7},{5,9},{6,10},{7,10}}
Cousin 8:
{{1,8},{1,9},{1,10},{2,5},{2,7},{2,8},{2,11},{3,4},{3,5},{3,6},{3,7},{3,8},{4,7},{4,9},{4,11},{5,6},{5,7},{5,9},{6,10},{6,11},{7,10}}
Cousin 9:
{{1,6},{1,7},{1,8},{1,9},{2,4},{2,5},{2,6},{2,7},{2,8},{3,4},{3,5},{3,6},{3,7},{3,8},{4,6},{4,7},{4,9},{5,6},{5,7},{5,9},{8,9}}
Cousin 10:
{{1,6},{1,7},{1,8},{1,9},{2,5},{2,7},{2,8},{2,10},{3,4},{3,7},{3,8},{3,11},{4,7},{4,9},{4,10},{5,7},{5,9},{5,11},{6,7},{6,10},{6,11}}
Cousin 11:
{{1,6},{1,7},{1,8},{1,9},{2,5},{2,7},{2,8},{2,10},{3,4},{3,6},{3,8},{3,11},{4,7},{4,9},{4,10},{5,6},{5,9},{5,11},{6,7},{6,10},{7,11}}
Cousin 12:
{{1,8},{1,9},{1,10},{2,8},{2,11},{2,12},{3,4},{3,5},{3,6},{3,7},{3,8},{4,7},{4,9},{4,11},{5,6},{5,9},{5,12},{6,10},{6,11},{7,10},{7,12}}
Cousin 13:
{{1,8},{1,9},{1,10},{2,5},{2,7},{2,8},{2,11},{3,4},{3,6},{3,8},{3,12},{4,7},{4,9},{4,11},{5,6},{5,9},{5,12},{6,10},{6,11},{7,10},{7,12}}
Cousin 14:
{{1,2},{1,4},{1,6},{2,5},{2,7},{2,8},{3,4},{3,5},{3,6},{3,7},{3,8},{4,7},{4,9},{5,6},{5,7},{5,9},{6,10},{7,10},{8,9},{8,10},{9,10}}
Cousin 15:
{{1,8},{1,9},{1,10},{2,8},{2,11},{2,12},{3,5},{3,6},{3,8},{3,13},{4,9},{4,11},{4,13},{5,6},{5,9},{5,12},{6,10},{6,11},{7,10},{7,12},{7,13}}
Cousin 16:
{{1,2},{1,5},{1,7},{2,8},{2,11},{3,4},{3,5},{3,6},{3,7},{3,8},{4,7},{4,9},{4,11},{5,6},{5,9},{6,10},{6,11},{7,10},{8,9},{8,10},{9,10}}
Cousin 17:
{{1,3},{1,5},{1,7},{2,5},{2,7},{2,8},{2,11},{3,4},{3,6},{3,8},{4,7},{4,9},{4,11},{5,6},{5,9},{6,10},{6,11},{7,10},{8,9},{8,10},{9,10}}
Cousin 18:
{{1,8},{1,9},{1,10},{2,8},{2,11},{2,12},{3,8},{3,13},{3,14},{4,9},{4,11},{4,13},{5,9},{5,12},{5,14},{6,10},{6,11},{6,14},{7,10},{7,12},{7,13}}
Cousin 19:
{{1,3},{1,4},{1,7},{2,8},{2,11},{2,12},{3,5},{3,6},{3,8},{4,9},{4,11},{5,6},{5,9},{5,12},{6,10},{6,11},{7,10},{7,12},{8,9},{8,10},{9,10}}
Cousin 20:
{{1,2},{1,5},{1,7},{1,8},{2,4},{2,6},{2,8},{3,4},{3,5},{3,6},{3,7},{3,8},{4,7},{4,9},{5,6},{5,9},{6,10},{7,10},{8,9},{8,10},{9,10}}
Diagram for the K_7 family: