A. O. Prishlyak

We define a $p$-graph and describe how it changes under isotopy of projections for classification of maps of $2$-manifolds into plane. The problem of graph implementation and maps classification are considered.

The present paper deals with the correspondence between Morse functions and flows on nonorientable surfaces. It is proved that for every Morse flow with an indexing of saddle points on a nonorientable surface there is a unique Morse function, up to a fiber equivalence, such that its gradient flow is trajectory equivalent to the initial flow, and the values of the function in the saddle points are ordered according to the indexing. The algorithm for constructing the Morse function from a Morse flow with an indexing is given. Reeb graphs and 3-graphs, which assign Morse functions and the corresponding Morse flows with the number of the saddle points less than $3$ are presented.

We give a topological classification of noncompact surfaces with any number of boundary components.