It is a class of search problems that can be shown to be total by an application of the handshaking lemma: any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number.
-bit binary strings, and the graph is represented by a polynomial-sized circuit that takes a vertex as input and outputs its neighbors.
(Note that this allows us to represent an exponentially-large graph on which we can efficiently perform local exploration.)
Suppose furthermore that a specific vertex (say the all-zeroes vector) has an odd number of neighbors.
This is because the corresponding problem that defines PPAD, known as END OF THE LINE, can be reduced (in a straightforward way) to the above search for an additional odd-degree vertex (essentially, just by ignoring the directions of the edges in END OF THE LINE).