On combinatorial invariant of pseudo-harmonic function defined on k-connected closed domain

Abstract

Let f be a pseudo-harmonic function defined on  k-connected oriented closed domain  D whose boundary consists of finitely many  closedJordancurves. We remind that this class of functions coincides with continuous functions which have finitely many number of critical points at interior and on boundary of domain.    

In [4] authors researched a case of  disk: for such functions a topological invariant is constructed, its main properties, the criterion of their topological equivalence and conditions of realization of some type of graphs as given invariant are proved.

In this paper, for case k>0 the combinatorial invariant G(f) of pseudo-harmonic function f is constructed that consists of the Reeb graphs of restriction of f on boundary of D and connected components such critical and semiregular levels which contain critical and boundary critical points.  According to a construction of G(f), it's a mixed pseudograph (graph with multiple edges and loops) with strict partial order on vertices which induced by values of f. There are two types of cycles in G(f). In particular, C-cycle (a simple cycle whose any pair of adjacent vertices are comparable) and L-cycle (a simple cycle whose any pair of adjacent vertices are noncomparable). Theorem of an invariant structure and a fact that a quantity of C-cycles of combinatorial invariant is same as a number of boundary curves of k-connected closed oriented domain are proved