Incidence structure

In mathematics, an incidence structure is a triple

$C=(P,L,I).\,$

where P is a set of "points", L is a set of "lines" and $I \subseteq P \times L$ is the incidence relation. The elements of I are called flags. If

$(p,l) \in I,$

we say that point p "lies on" line l. One may concretely have L be a set of subsets of P, and have incidence I be containment ($(p,l) \in I$ if and only if $p \in l$), but one may also work more abstractly.

Incidence structures generalize planes (such as affine, projective, and Möbius planes) in their axiomatic definitions, as the terminology indicates. The higher-dimensional analog is called an incidence geometry.

Comparison with other structures

A figure may look like a graph, but in a graph an edge has just two ends (beyond a vertex a new edge starts), while a line in an incidence structure can be incident to more points.

An incidence structure has no concept of a point being in between two other points; the order of points on a line is undefined. Compare with ordered geometry, which does have a notion of betweenness.

Dual structure

If we interchange the role of "points" and "lines" in

C=(P,L,I)

the dual structure

C*=(L,P,I*)

is obtained, where I* is the inverse relation of I. Clearly

C**=C.

This is an abstract version of projective duality.

A structure C that is isomorphic to its dual C* is called self-dual.

Correspondence with hypergraphs

Seven points are elements of seven lines in the Fano plane

Each hypergraph or set system can be regarded as an incidence structure in which the universal set plays the role of "points", the corresponding family of sets plays the role of "lines" and the incidence relation is set membership "∈". Conversely, every incidence structure can be viewed as a hypergraph.

Example: Fano plane

In particular, let

P = {1, 2, 3, 4, 5, 6, 7},

L = { {1,2,3}, {1,4,5}, {1,6,7}, {2,4,6}, {2,5,7}, {3,4,7}, {3,5,6} }.

The corresponding incidence structure is called the Fano plane.

The lines are precisely the subsets of the points that consist of three points whose labels add up to zero using nim addition.

Geometric representation

Incidence structures can be modelled by points and curves in the Euclidean plane with usual geometric incidence. Some incidence structures admit representation by points and lines. The Fano plane is not one of them since it needs at least one curve.

Levi graph of an incidence structure

Heawood graph with labeling

Each incidence structure C corresponds to a bipartite graph called Levi graph or incidence graph with a given black and white vertex coloring where black vertices correspond to points and white vertices correspond to lines of C and the edges correspond to flags.

Example: Heawood graph

For instance, the Levi graph of the Fano plane is the Heawood graph. Since the Heawood graph is connected and vertex-transitive, there exists an automorphism (such as the one defined by a reflection about the vertical axis in the above figure) interchanging black and white vertices. This, in turn, implies that the Fano plane is self-dual.

See also

• Binary relation
• Incidence matrix
• Incidence (geometry)
• Projective configuration
• Pappus configuration
• Hypergraph

References

• CRC Press (2000). Handbook of discrete and combinatorial mathematics, (Chapter 12.2), ISBN 0-8493-0149-1