In memory of
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I call the teams who are opponents of either team in a pair with no common opponents "glued" by the pair. Team A becomes an opponents' opponent for each of team B's opponents and team B becomes an Os O for each of team A's opponents, and without this A-B pair they would be no better than an Os-Os-O, except for those opponents of A who play an opponent of B but not B, and vice versa for those opponents of B.
To take the deeper look we find all of the subsets of the games graph where every team in the subset plays every other team in it.
As you can see from the table at the right, there are 24,106 subsets of the D1 schedule graph wherein every team plays every other team in the subset, but only 1800 that are not members of a larger subset. In general we need only look at the unique subsets for those that glue the games graph into a connected whole.
Notice that there are no round-robins with more than seven teams that have representatives from more than one conference. The 13 largest are all determined by conference schedules:
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As one would expect, multiconference round robins in general contribute more to how well-connected the field is. Not all are equally efficient in that regard, though. We can measure efficiency a couple of ways. The number of Opponents' Opponent pairs contributed by a round robin of size N (Cont) is
Considering only the round robins involving three or more teams not all from the same conference, here are those that contribute at least 200 or at most 60:(Glued − Dupe) × (N-1)
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You can get a feel for the skeleton of the games graph by browsing 2016 Scheduled Round Robins where you will find all 1704 sorted in decreasing per-team contribution to O-Os for all round robins.
The interconference game-count matrix I included last time is symmetrical - the same number appears in the (confA,confB) position and (confB,confA). You can't tell from the number of games how many teams from confA play at least one team in confB, which usually is not the same as the number of teams in confB that play teams in confA. Here is a breakdown that does:
| AE | AAC | A10 | ACC | ASUN | B10 | B12 | BigE | BigS | BigW | CAA | CUSA | Hor | Ind | Ivy | MAAC | MAC | MEAC | MVC | MW | NE | OVC | P12 | Pat | SEC | SoCon | SLC | SWAC | Summ | SBC | WCC | WAC | ||
| AE | ∗ | 2 | 6 | 3 | 3 | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 0 | 5 | 5 | 6 | 1 | 3 | 0 | 0 | 6 | 0 | 0 | 5 | 2 | 3 | 2 | 0 | 1 | 1 | 1 | 1 | AE |
| AAC | 1 | ∗ | 4 | 5 | 3 | 5 | 2 | 2 | 4 | 1 | 2 | 8 | 1 | 0 | 3 | 4 | 2 | 2 | 4 | 0 | 2 | 2 | 2 | 1 | 7 | 2 | 4 | 4 | 0 | 4 | 2 | 1 | AAC |
| A10 | 7 | 5 | ∗ | 5 | 4 | 8 | 1 | 10 | 10 | 3 | 10 | 2 | 1 | 2 | 7 | 11 | 5 | 5 | 3 | 1 | 8 | 3 | 0 | 7 | 3 | 6 | 0 | 0 | 3 | 2 | 2 | 2 | A10 |
| ACC | 5 | 8 | 7 | ∗ | 3 | 7 | 3 | 6 | 9 | 0 | 8 | 6 | 4 | 0 | 4 | 8 | 8 | 5 | 3 | 0 | 3 | 2 | 4 | 2 | 8 | 8 | 2 | 3 | 1 | 4 | 1 | 1 | ACC |
| ASUN | 3 | 3 | 4 | 4 | ∗ | 3 | 0 | 4 | 3 | 0 | 3 | 5 | 0 | 1 | 3 | 4 | 4 | 5 | 1 | 0 | 2 | 2 | 0 | 3 | 6 | 5 | 0 | 1 | 0 | 3 | 0 | 0 | ASUN |
| B10 | 1 | 6 | 7 | 8 | 3 | ∗ | 7 | 7 | 6 | 5 | 4 | 4 | 6 | 1 | 3 | 4 | 10 | 1 | 9 | 4 | 3 | 2 | 5 | 2 | 5 | 4 | 3 | 0 | 5 | 3 | 3 | 5 | B10 |
| B12 | 1 | 4 | 1 | 2 | 0 | 5 | ∗ | 2 | 1 | 4 | 2 | 4 | 3 | 0 | 0 | 1 | 1 | 0 | 6 | 4 | 0 | 2 | 7 | 0 | 4 | 4 | 7 | 4 | 5 | 7 | 3 | 7 | B12 |
| BigE | 4 | 2 | 4 | 4 | 4 | 5 | 1 | ∗ | 1 | 0 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 5 | 2 | 3 | 3 | 1 | 3 | 2 | 3 | 0 | 0 | 2 | 0 | 1 | 2 | BigE |
| BigS | 1 | 5 | 8 | 11 | 7 | 5 | 2 | 1 | ∗ | 0 | 8 | 5 | 1 | 0 | 0 | 4 | 7 | 8 | 2 | 0 | 3 | 2 | 0 | 2 | 6 | 10 | 0 | 1 | 0 | 5 | 0 | 0 | BigS |
| BigW | 1 | 2 | 2 | 0 | 0 | 7 | 4 | 0 | 0 | ∗ | 2 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 2 | 8 | 0 | 0 | 9 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 9 | 8 | BigW |
| CAA | 3 | 4 | 9 | 8 | 3 | 6 | 2 | 2 | 7 | 2 | ∗ | 2 | 1 | 1 | 2 | 4 | 3 | 6 | 0 | 0 | 4 | 1 | 1 | 5 | 2 | 4 | 1 | 0 | 1 | 2 | 1 | 1 | CAA |
| CUSA | 1 | 7 | 2 | 6 | 6 | 5 | 4 | 2 | 3 | 1 | 2 | ∗ | 2 | 0 | 2 | 3 | 5 | 4 | 3 | 0 | 1 | 4 | 1 | 2 | 9 | 4 | 4 | 2 | 1 | 8 | 2 | 0 | CUSA |
| Hor | 0 | 1 | 3 | 4 | 0 | 6 | 2 | 2 | 1 | 0 | 1 | 2 | ∗ | 0 | 0 | 1 | 7 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 4 | 1 | 1 | 3 | 1 | 3 | 1 | 4 | Hor |
| Ind | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | ∗ | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | Ind |
| Ivy | 6 | 5 | 7 | 4 | 3 | 3 | 0 | 5 | 0 | 1 | 2 | 2 | 0 | 2 | ∗ | 5 | 0 | 0 | 2 | 0 | 1 | 0 | 0 | 8 | 4 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | Ivy |
| MAAC | 8 | 4 | 11 | 6 | 6 | 6 | 1 | 3 | 5 | 0 | 5 | 2 | 1 | 3 | 6 | ∗ | 2 | 4 | 1 | 1 | 9 | 1 | 0 | 7 | 1 | 3 | 0 | 0 | 0 | 1 | 1 | 0 | MAAC |
| MAC | 1 | 3 | 10 | 9 | 6 | 10 | 1 | 4 | 5 | 0 | 2 | 5 | 10 | 0 | 0 | 3 | ∗ | 3 | 5 | 1 | 1 | 5 | 2 | 0 | 6 | 3 | 0 | 0 | 4 | 2 | 1 | 3 | MAC |
| MEAC | 4 | 2 | 5 | 7 | 6 | 1 | 0 | 2 | 5 | 0 | 4 | 4 | 0 | 1 | 0 | 4 | 2 | ∗ | 0 | 0 | 4 | 3 | 0 | 5 | 2 | 3 | 0 | 2 | 0 | 4 | 0 | 0 | MEAC |
| MVC | 0 | 4 | 5 | 2 | 1 | 8 | 4 | 5 | 2 | 1 | 0 | 3 | 2 | 0 | 2 | 1 | 6 | 0 | ∗ | 2 | 1 | 6 | 1 | 1 | 1 | 1 | 5 | 3 | 6 | 5 | 1 | 2 | MVC |
| MW | 0 | 0 | 1 | 0 | 0 | 5 | 4 | 3 | 0 | 7 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | ∗ | 1 | 1 | 5 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 6 | 6 | MW |
| NE | 5 | 3 | 5 | 3 | 2 | 2 | 0 | 4 | 4 | 0 | 5 | 1 | 0 | 2 | 1 | 5 | 1 | 4 | 1 | 1 | ∗ | 0 | 0 | 3 | 0 | 0 | 2 | 1 | 0 | 1 | 0 | 0 | NE |
| OVC | 0 | 6 | 6 | 3 | 7 | 2 | 2 | 3 | 2 | 0 | 1 | 6 | 2 | 0 | 0 | 1 | 7 | 3 | 8 | 1 | 0 | ∗ | 0 | 2 | 9 | 2 | 3 | 7 | 6 | 5 | 0 | 0 | OVC |
| P12 | 0 | 2 | 0 | 4 | 0 | 4 | 7 | 1 | 0 | 9 | 1 | 1 | 0 | 0 | 0 | 0 | 3 | 0 | 1 | 4 | 0 | 0 | ∗ | 0 | 4 | 0 | 4 | 0 | 0 | 1 | 11 | 7 | P12 |
| Pat | 5 | 1 | 6 | 1 | 3 | 3 | 0 | 4 | 1 | 1 | 5 | 2 | 0 | 2 | 6 | 4 | 0 | 2 | 2 | 1 | 4 | 2 | 0 | ∗ | 0 | 3 | 0 | 0 | 1 | 1 | 0 | 0 | Pat |
| SEC | 2 | 10 | 3 | 9 | 5 | 6 | 4 | 3 | 5 | 1 | 2 | 13 | 4 | 0 | 3 | 1 | 7 | 2 | 2 | 1 | 0 | 7 | 3 | 0 | ∗ | 5 | 6 | 9 | 3 | 10 | 4 | 4 | SEC |
| SoCon | 3 | 2 | 6 | 8 | 6 | 4 | 4 | 4 | 8 | 0 | 7 | 3 | 1 | 0 | 1 | 4 | 3 | 4 | 1 | 0 | 0 | 2 | 0 | 4 | 7 | ∗ | 0 | 1 | 0 | 4 | 1 | 0 | SoCon |
| SLC | 2 | 7 | 0 | 2 | 0 | 3 | 8 | 0 | 0 | 1 | 1 | 9 | 1 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 3 | 3 | 6 | 0 | 13 | 0 | ∗ | 11 | 1 | 11 | 2 | 5 | SLC |
| SWAC | 0 | 7 | 0 | 3 | 2 | 0 | 2 | 0 | 1 | 0 | 0 | 5 | 3 | 0 | 0 | 0 | 0 | 3 | 2 | 0 | 1 | 5 | 0 | 0 | 8 | 2 | 9 | ∗ | 2 | 6 | 1 | 1 | SWAC |
| Summ | 1 | 0 | 3 | 1 | 0 | 5 | 2 | 2 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 3 | 0 | 3 | 1 | 0 | 4 | 0 | 1 | 3 | 0 | 1 | 2 | ∗ | 3 | 1 | 4 | Summ |
| SBC | 1 | 5 | 2 | 3 | 2 | 3 | 5 | 0 | 2 | 0 | 2 | 8 | 3 | 0 | 0 | 1 | 2 | 2 | 5 | 1 | 1 | 4 | 1 | 1 | 10 | 4 | 8 | 6 | 2 | ∗ | 0 | 1 | SBC |
| WCC | 2 | 1 | 2 | 2 | 0 | 4 | 4 | 1 | 0 | 9 | 1 | 2 | 2 | 0 | 2 | 1 | 1 | 0 | 1 | 6 | 0 | 0 | 10 | 0 | 4 | 1 | 2 | 1 | 1 | 0 | ∗ | 9 | WCC |
| WAC | 1 | 1 | 3 | 1 | 0 | 5 | 6 | 2 | 0 | 8 | 1 | 0 | 2 | 0 | 0 | 0 | 3 | 0 | 3 | 5 | 0 | 0 | 8 | 0 | 3 | 0 | 4 | 1 | 5 | 1 | 7 | ∗ | WAC |
| AE | AAC | A10 | ACC | ASUN | B10 | B12 | BigE | BigS | BigW | CAA | CUSA | Hor | Ind | Ivy | MAAC | MAC | MEAC | MVC | MW | NE | OVC | P12 | Pat | SEC | SoCon | SLC | SWAC | Summ | SBC | WCC | WAC |
In memory of