Order dimension explained

In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see .

Formal definition

The order dimension of a poset

is the least integer

for which there exists a family

lR=(<_1,...,<_t)

of linear extensions of

so that, for every

and

precedes

if and only if it precedes

in all of the linear extensions, if any such

exists. In other words, that the intersection of those linear extensions equals

. That is,

	t
P=caplR=cap
	i=1

<_i.

An alternative definition of order dimension is the minimal number of total orders such that P embeds into their product with componentwise ordering i.e.

x\leqy

if and only if

x_i\leqy_i

for all i .

Realizers

A family

lR=(<_1,...,<_t)

of total orders on

is called a realizer of a poset P=(X,<_P)

if

<_P=caplR

which is to say that for any

and

, x<_Py

precisely when x<₁y,x<₂y

,..., x<_ty

.Thus, an equivalent definition of the dimension of a poset

is "the least cardinality of a realizer of

It can be shown that any nonempty family

of linear extensions is a realizer of a finite partially ordered set

if and only if, for every critical pair

(x,y)

of,

x<_iy

for some order

<_i

Example

Let n be a positive integer, and let P be the partial order on the elements a_i and b_i (for 1 ≤ i ≤ n) in which a_i ≤ b_j whenever i ≠ j, but no other pairs are comparable. In particular, a_i and b_i are incomparable in P; P can be viewed as an oriented form of a crown graph. The illustration shows an ordering of this type for n = 4.

Then, for each i, any realizer must contain a linear order that begins with all the a_j except a_i (in some order), then includes b_i, then a_i, and ends with all the remaining b_j. This is so because if there were a realizer that didn't include such an order, then the intersection of that realizer's orders would have a_i preceding b_i, which would contradict the incomparability of a_i and b_i in P. And conversely, any family of linear orders that includes one order of this type for each i has P as its intersection. Thus, P has dimension exactly n. In fact, P is known as the standard example of a poset of dimension n, and is usually denoted by S_n.

Order dimension two

The partial orders with order dimension two may be characterized as the partial orders whose comparability graph is the complement of the comparability graph of a different partial order . That is, P is a partial order with order dimension two if and only if there exists a partial order Q on the same set of elements, such that every pair x, y of distinct elements is comparable in exactly one of these two partial orders. If P is realized by two linear extensions, then partial order Q complementary to P may be realized by reversing one of the two linear extensions. Therefore, the comparability graphs of the partial orders of dimension two are exactly the permutation graphs, graphs that are both themselves comparability graphs and complementary to comparability graphs.

The partial orders of order dimension two include the series-parallel partial orders . They are exactly the partial orders whose Hasse diagrams have dominance drawings, which can be obtained by using the positions in the two permutations of a realizer as Cartesian coordinates.

Computational complexity

It is possible to determine in polynomial time whether a given finite partially ordered set has order dimension at most two, for instance, by testing whether the comparability graph of the partial order is a permutation graph. However, for any k ≥ 3, it is NP-complete to test whether the order dimension is at most k .

Order dimension of incidence posets of graphs

The incidence poset of any undirected graph G has the vertices and edges of G as its elements; in this poset, x ≤ y if either x = y or x is a vertex, y is an edge, and x is an endpoint of y. Certain kinds of graphs may be characterized by the order dimensions of their incidence posets: a graph is a path graph if and only if the order dimension of its incidence poset is at most two, and according to Schnyder's theorem it is a planar graph if and only if the order dimension of its incidence poset is at most three .

For a complete graph on n vertices, the order dimension of the incidence poset is

\Theta(loglogn)

. It follows that all simple n-vertex graphs have incidence posets with order dimension

O(loglogn)

Order dimension of planar maps

Schnyder's theorem can be generalized for the vertex-edge-face incidence poset, denoted by

P_M

, of a planar map

. As before

x<y

is a vertex,

an edge and

an endpoint of

. On the other hand

x<y

will be true if

is an edge,

a face and

incident to

. Thus, the incidence poset is defined by the natural incidence relations between vertices, edges, and faces.

It was shown by that the order dimension of the vertex-edge-face incidence poset of a planar map (planar graph with fixed plane embedding) is at most four.

later proved in "The order dimension of planar maps revisited " that

dim(bf{split}(P_M))\leq4

For an order

P=(X,<)

the

bf{split}

is defined as

bf{split}(P)=(X'\cupX'',<_s)

, whereas

and

X''

are two copies on

. The order

<_s

is given by:

x'<_sx'' \forallx\inX

x'<_sy''

if and only if

x<y

It is to be noted, that

dim(P)\leqdim(bf{split}(P))\leqdim(P)+1

Felsner's proof of that the order dimension of the split incidence poset is at most four constructs a grid intersection representation of the respective planar map in which incidences correspond to intersections of axis-parallel segments. This representation makes it possible to construct four linear extensions whose intersection realizes the split incidence poset, establishing the dimension bound of four.

k-dimension and 2-dimension

A generalization of dimension is the notion of k-dimension (written

rm{dim}_k

) which is the minimal number of chains of length at most k in whose product the partial order can be embedded. In particular, the 2-dimension of an order can be seen as the size of the smallest set such that the order embeds in the inclusion order on this set.

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