In mathematics, particularly in combinatorics, an incidence structure is a triple

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

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

$(p,l)\; \backslash in\; I,$

we say that "point" $p\backslash ,\backslash !$ lies on "line" $l\backslash ,\backslash !$.

Comparison with other structures

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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.

Dual structure

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

$C=(P,L,I),\backslash ,\backslash !$

the dual structure

$C^*=(L,P,I^*)\backslash ,\backslash !$

is obtained. Clearly

$C^\{**\}=C.\backslash ,\backslash !$

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

Hypergraphs as incidence structures

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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 given by set membership "∈".

Example: Fano plane

In particular, let

P = {1,2,3,4,5,6,7},

L = 

The corresponding incidence structure is called the Fano plane.

Geometric representation

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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. The Fano plane can, however, be modelled in Euclidean 3-space by a set of seven partitions of a 2×2×2 cube; see The Eightfold Cube. For other models of incidence structures, see The Diamond Theorem and Finite-Geometry Models.

Incidence structure and its Levi graph

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To each incidence structure $C$ we may associate 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, it follows that 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

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• Binary relation
• Incidence (geometry)
• Pappus configuration

Set families | Combinatorics | Geometry | Incidence geometry