category of paths on a graph
Let be an undirected graph. Denote the set of vertices of by “” and denote the set of edges of by “”.
A path of the graph is an ordered tuplet of vertices such that, for all between and , there exists an edge connecting and . As a special case, we allow trivial paths which consist of a single vertex — soon we will see that these in fact play an important role as identity elements in our category.
In our category, the vertices of the graph will be the objects and the morphisms will be paths; given two of these objects and , we set to be the set of all paths such that and . Given an object , we set , the trivial path mentioned above.
(Remember that .) To have a bona fide category, we need to check that this choice satisfies the defining properties (A1 - A3 in the entry http://planetmath.org/node/965category). This is rather easily verified.
A1: Given a morphism , it can only belong to if and , hence unless and .
A2: Suppose that we have four objects and three morphisms, , , and . Then, by the definition of the operation given above,
Since these two quantities are equal, the operation is associative.
A3: It is easy enough to check that paths with a single vertex act as identity elements:
It is also possible to consider the equivalence class of paths modulo retracing. To introduce this category, we first define a binary relation on the class of paths as follows: Let and be any two paths such that the right endpoint of is the same as the left endpoint of , i.e. and for some vertices of our graph. Let be any vertex which shares an edge with . Then we set .
Let be the smallest equivalence relations which contains . We will call this equivalence relation retracing.
As defined above, it may not intuitively obvious what this equivalence amounts to. To this end, we may consider a different description. Define the reversal of a path to be the path obtained by reversing the order of the vertices traversed:
Then we may show that two paths are equivalent under retracing if they may both be obtained from a third path by inserting terms of the form . In symbols, we claim that if there exists an integer and paths such that
This characterization explains the choice of the term “retracing” — we do not change the equivalence class of the path if we happen to wander off somewhere in the course of following the path but then backtrack and pick the path up again where we left off on our digression.
Rather than presenting a detailed formal proof, we will sketch how the two definitions may be shown to be equivalent.
|Title||category of paths on a graph|
|Date of creation||2013-03-22 16:45:54|
|Last modified on||2013-03-22 16:45:54|
|Last modified by||rspuzio (6075)|