For every point and every line there exists a unique point on which is nearest to.
Note that the distance are measured in the collinearity graph of points, i.e., the graph formed by taking points as vertices and joining a pair of vertices if they are incident with a common line. We can also give an alternate graph theoreticdefinition, a near 2d-gon is a connected graph of finite diameter d with the property that for every vertex x and every maximal cliqueM there exists a unique vertex x' in M nearest to x. The maximal cliques of such a graph correspond to the lines in the incidence structure definition. A near 0-gon is a single point while a near 2-gon is just a single line, i.e., a complete graph. A near quadrangle is same as a generalized quadrangle. In fact, it can be shown that every generalized 2d-gon is a near 2d-gon that satisfies the following two additional conditions:
Every point is incident with at least two lines.
For every two points x, y at distance i < d, there exists a unique neighbour of y at distance i − 1 from x.
A near polygon is called dense if every line is incident with at least three points and if every two points at distance two have at least two common neighbours. It is said to have order if every line is incident with precisely s + 1 points and every point is incident with precisely t + 1 lines. Dense near polygons have a rich theory and several classes of them have been completely classified.
Examples
All connected bipartite graphs are near polygons. In fact, any near polygon that has precisely two points per line must be a connected bipartite graph.
All finite generalized polygons except the projective planes.
The Hall–Janko near octagon, also known as the Cohen-Tits near octagon associated with the Hall–Janko group. It can be constructed by choosing the conjugacy class of 315 central involutions of the Hall-Janko group as points and lines as three element subsets whenever x and y commute.
Take the partitions of into n + 1 2-subsets as points and the partitions into n − 1 2-subsets and one 4-subset as lines. A point is incident to a line if as a partition it is a refinement of the line. This gives us a near 2n-gon with three points on each line, usually denoted Hn. Its full automorphism group is the symmetric groupS2n+2.
Regular near polygons
A finite near -gon S is called regular if it has an order and if there exist constants, such that for every two points and at distance, there are precisely lines through containing a point at distance from. It turns out that regular near -gons are precisely those near -gons whose point graph is a distance-regular graph. A generalized -gon of order is a regular near -gon with parameters
