There is no general characterization of graphs where alpha equals max_even_minus_even_horizontal. This would be really useful – as the bound is very good – at least for small graphs:

Equality between these invariants holds for:

- 1 of 1 connected graphs on 1 vertex
- 1 of 1 connected graphs on 2 vertices
- 2 of 2 connected graphs on 3 vertices
- 6 of 6 connected graphs on 4 vertices
- 21 of 21 connected graphs on 5 vertices
- 109 of 112 connected graphs on 6 vertices
- 796 of 853 connected graphs on 7 vertices
- 9,572 of 11,117 connected graphs on 8 vertices
- 195,096 of 261,080 connected graphs on 9 vertices
- 698,0595 of 11,716,571 connected graphs on 10 vertices

Is there a way to use the computer to help find a characterization of equality???

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