Topological equivalence of the pseudoharmonic functions defined on the D².
by
Iryna Yurchuk
Institiute of mathematics of NASU
Coauthors: Yevgen Polulyakh

We get the topological classification of the pseudoharmonic functions defined on the disk in term of their invariant. Necessary and sufficient conditions for a finite connected graph with a strict partial order on vertices to be a combinatorial invariant of the pseudoharmonic functions are obtained.

Let us remind that a pseudoharmonic function defined on D² ⊂ C is a continuous function such that f|_∂D² has a finite number of extrema and has only saddle critical points in the interior of D² (see [1]). The invariant of such functions was obtained in [2]. It is a combinatorial diagram P(f) that is a finite connected graph with a strict partial order on its vertices. We list its properties:

A1) there exists a unique Cr-subgraph q(f) of P(f) such that it is a simple cycle and every pair of its adjacent vertices is comparable;

A2) Cl(P(f)\q(f))=∪_i Y_i, Y_j∩Y_i=∅ where i ≠ j and every Y_i is a tree such that any two vertices v', v" of Y_i are non comparable;

A3) there is an embedding y:P(f)→ D² such that y(P(f)) ⊂ D², y(q(f))=∂D² and y(P(f)\q(f)) ⊂ Int D².

A4) D²\y(P(f))=∪q_i where q_j∩q_i=∅, i ≠ j, Cl(q_i) is homeomorphic to a disk and \partialCl(q_i) contains one or two arcs of ∂D².

Let f, g:D²→ R be pseudoharmonic functions. Two functions f and g are called topologically equivalent if there exist orientation preserving homeomorphisms h₁:D²→ D² and h₂:R→ R such that g=h₂ ○f ○h₁.

Theorem 1. Two pseudoharmonic functions f and g are topologically equivalent if and only if there exists an isomorphism j:P(f)→ P(g) preserving the strict partial orders on them.

Now we will consider the inverse problem of the realization of some finite graph as a combinatorial invariant of a pseudoharmonic function.

Let G be a finite connected graph with a strict partial order on its vertices. In order to be a combinatorial diagram of some pseudogarmonic function G should comply with properties A1-A4 and we add some extra one's.

Cr-cycle of G is a subgraph g ⊂ G such that it is the simple cycle and every its adjacent pair of vertices is comparable.

Let G satisfies following conditions:

B1) a graph G has a unique Cr-cycle g;

B2) if we throw away all edges which belong to g from G we obtain a forest F=∪Y_i where any Y_i is a tree such that its all terminal vertices belong to g.

A graph G which satisfies to B1)-B2) is called D-planar if there is an embedding j: G → D² such that j(g)=∂D², j(G\g) ⊂ Int D².

There is a cyclic order on g generated by an orientation of the disk. For every i this cyclic order generates cyclic order on a set V_i^* = g∩V(Y_i) (here V(Y_i) is a set of vertices of a tree Y_i).

Let T be a finite tree with a fixed cyclically ordered subset of vertices V^* ⊇ V_ter where V_ter is a set of terminal vertices of T. A tree T is called D-planar if there exists an embedding j: T → R² that satisfies following conditions: j(T) ⊆ D², j(T) ∩∂D² = j(V^*) and if |V^*| ≥ 3 then a cyclic order j(C) on a set j(V^*) coincides with a cyclic order generated by an orientation of ∂D² ≅ S¹.

Theorem 2. If |V^*| = 2 then a tree T is D-planar.

If |V^*| ≥ 3 then the D-planarity of a tree T is equivalent to the following condition being satisfied: for every edge e there are exactly two paths such that they pass through an edge e and connect adjacent vertices of V^*.

Theorem 3. A graph G satisfying B1)-B2) is D-planar if and only if every tree Y_i with a subset of vertices V_i^* that has a cyclic order defined by g is D-planar and for any indices m ≠ n a set V_n^* belongs to a unique connected component of g\V^*_m.

Let consider any two vertices v₁, v₂ ∈ V^*_i of some subgraph Y_i of G satisfying B1)-B2). The set g\{v₁, v₂} consists of a disjoint union of two arcs g₁ and g₂.

A pair of vertices v₁, v₂ ∈ V_i^* is called a boundary pair if either g₁ or g₂ does not contain any vertices of V^*_i and at least one vertex of V^*\V^*_i belongs to it.

By w(v₁, v₂) and a_i we denote a boundary pair and a set g_k from definition 3, respectively. It is clear that for every vertex v_s of a boundary pair w(v₁, v₂) there exists an adjacent vertex [v\tilde]_s such that [v\tilde]_s ∈ a_i where s ∈ 1, 2.

A graph G is called a D-graph if it satisfies B1), B2) and the following conditions:

B3) if for some vertices of G it is true that v < v' (v > v') and v', v" ∈ Y_i ⊆ Cl(G \g) then v < v" (v > v");

B4) for any vertex v ∈ G \g it is true that deg(v)=2s ≥ 4;

B5) for any boundary pair w(v₁, v₂) ∈ V_i^* a pair of adjacent vertices [v\tilde]₁, [v\tilde]₂ ∈ a_i belongs to a unique set V^*_k ⊂ V^*\V_i^*;

B6) for any vertex v of g and its adjacent vertices v₁ and v₂ such that v₁, v₂ ∈ g it holds that: if deg(v)=2 then deg(v₁) > 2, deg(v₂) > 2 and there exists a unique index i such that v₁, v₂ ∈ Y_i; if deg(v)=2s > 2 (deg(v)=2s+1) then v₁ < v > v₂ or v₁ > v < v₂(v₁ < v < v₂ or v₁ > v > v₂);

B7) G is D - planar.

Theorem 4. If a graph is a combinatorial diagram of a pseudoharmonic function f then G is D - graph.

If a graph G is D - graph then a strict partial order on V(G) can be extended to one so that a graph G with new partial order on a set of vertices will be isomorphic to combinatorial diagram of some pseudoharmonic function f. A strict partial order of a graph G is the same as a strict partial order of a combinatorial diagram P(f) of a pseudoharmonic function f if and only if G satisfies:

B8) if vertices v', v" are non comparable then from v > v' follows v > v" where v ∈ G, v ≠ v', v ≠ v".

References

[1] M. Morse, The topology of pseudo-harmonic functions// Duke Math.J.-1946.-13.-P.21-42.

[2] I. Yurchuk, Topological equivalence of functions from F(D²) // Zb.Inst.Math NAS Ukraine (in Ukrainian), 2006.- T.3, No 3. - P. 474-486.

Date received: April 29, 2008