Note that I do not work in combinatorics, and so this question might be a bit naive. The question is inspired by some structures that arise in my research within representation theory.
Recall that an incidence structure consists of a triple $(P,L,I)$ where $P$ is a set of points, $L$ is a set of lines and $I$ is a relation between them.
The dual of an incidence structure $(P,L,I)$ is $(L,P,I)$, i.e. we interchange the roles of points and lines but keep the same incidence relation.
From an incidence structure $(P,L,I)$ we can get a bipartite graph, its incidence graph, by setting $G = (P \cup L, E)$ with $(p,l) \in E$ whenever $p$ is incident with $l$ for $p \in P$ and $l \in L$. Similarly, any bipartite graph $G$ yields an incidence structure. The complement of an incidence structure is that corresponding to the bipartite complement of its incidence graph (i.e. the complement of the incidence graph in the appropriate complete bipartite graph).
Finally, call an incidence structure connected if its incidence graph is connected.
I am interested in a class of incidences structures that is closed under duals and complements, is connected and satisfies that each point is incident with at least two lines, and that for the set of lines incident with a point, there is some point $p$ such that some subset of those lines are not incident with $p$. Moreover, for every point $p$ there is no other point incident with all the lines $p$ is incident with.
Of course, since such a class is closed under duals, the latter conditions must hold with "point" and "line" interchanged everywhere, too. Similarly, because of the closure under complements, there are more conditions that need to be satisfied derived from the ones already given.
There may be more conditions that we have not yet been able to determine, and here are some examples of structures that work and don’t work.
The class should include as examples the following:
- the Fano plane;
- the generalized quadrangles corresponding to the 2x2 and 3x3 grids, i.e. GQ(1,1) and GQ(2,1);
- the smallest Benz plane (i.e. the trivial 3-(5,3,1) design given by all subsets of cardinality 3 of a 5 element set); and
- all of their complements and duals.
Non-examples should include:
- the projective plane over GF(3);
- the affine plane over GF(2);
- the generalized hexagon GH(1,2); and
- the Doily GQ(2,2).
I was wondering if incidence structures satisfying these or similar properties were studied in some context, and if the examples and non-examples that I give are consistent with some already known class of structures. I should add that closure under complements is observed in examples, but we have not been able to show that it must hold.
