In geometry, a truncated projective plane (TPP), also known as a dual affine plane, is a special kind of a hypergraph or geometric configuration that is constructed in the following way.
[1][2] These objects have been studied in many different settings, often independent of one another, and so, many terminologies have been developed.
It is a tripartite hypergraph with sides {1,6},{2,5},{3,4} (which are exactly the neighbors of the removed vertex 7).
A finite projective plane of order n has n + 1 points on every line (n + 1 = r in the hypergraph description).
These sets are the analogs of classes of parallel lines in an affine plane, and some authors refer to the points in a partition piece as parallel points in keeping with the dual nature of the structure.
It is known that the projective plane of order r-1 exists whenever r-1 is a prime power; hence the same is true for the TPP.
The finite projective plane of order r-1 contains r2-r+1 vertices and r2-r+1 edges; hence the TPP of order r-1 contains r2-r vertices and r2-2r+1 edges.
.On the other hand, covering all edges of the TPP requires all r-1 vertices of one of the parts.
[5][1][6] The minimum fractional vertex-cover size of the TPP is r-1 too: assigning a weight of 1/r to each vertex (which is a vertex-cover since each hyperedge contains r vertices) yields a fractional cover of size (r2-r)/r=r-1.
.Note that the above fractional matching is perfect, since its size equals the number of vertices in each part of the r-partite hypergraph.
[10] Since they are not pairwise balanced designs (PBDs), they have not been studied extensively from the design-theoretic viewpoint.
According to Dembowski (1968, p. 5), the term "tactical configuration" appears to be due to E. H. Moore in 1896.