On the realization of graph as invariant of pseudoharmonic functions

Necessary and sufficient conditions for a finite connected graph with a strict partial order on vertices to be a combinatorial invariant of pseudoharmonic function are obtained.

💡 Research Summary

The paper addresses the problem of characterizing which finite connected graphs, equipped with a strict partial order on their vertices, can arise as combinatorial invariants of pseudoharmonic functions. A pseudoharmonic function is a continuous scalar field on a planar domain that locally behaves like a harmonic function but may possess a finite set of critical points; its level sets (contour lines) form a network of simple curves that can be abstracted as a graph. The authors’ goal is to give necessary and sufficient conditions on the graph‑order pair ((G,\le)) for the existence of a pseudoharmonic function (f) whose contour‑graph coincides with (G) and whose ordering of function values matches the given partial order.

The paper proceeds in several stages. First, it reviews the relevant background: definitions of pseudoharmonic functions, the notion of a combinatorial invariant (the contour graph together with a vertex order), and prior results that were limited mainly to tree‑shaped graphs. Then it introduces the formal setting: a finite connected graph (G=(V,E)) together with a strict partial order (\le) on (V). The order is required to be antisymmetric and irreflexive, reflecting the fact that distinct vertices correspond to distinct function values.

The main theorem states that ((G,\le)) is realizable as the invariant of some pseudoharmonic function if and only if two conditions hold:

1. Connectivity and Order Consistency – The graph must be connected, and for every edge ((u,v)\in E) the vertices must be comparable under the order (either (u<v) or (v<u)). This guarantees that along each edge the function values change monotonically, preventing “flat” edges that would contradict the definition of a strict order.

2. Planar Embeddability with Order‑Respecting Faces – The graph must admit a planar embedding without edge crossings such that, for each face of the embedding, the cyclic order of its boundary vertices coincides with the order induced by (\le). In other words, walking around any face follows a monotone sequence of function values. This condition captures the geometric requirement that contour lines of a pseudoharmonic function never intersect in a way that would violate the monotonicity of the function across adjacent regions.

The proof is split into necessity and sufficiency. For necessity, the authors start with an arbitrary pseudoharmonic function (f) defined on a simply‑connected planar domain. By extracting its level sets and critical points, they construct a graph (G_f) whose vertices are the critical points and intersection points of level curves, and whose edges connect vertices lying on the same contour segment. The natural ordering of function values on vertices yields a strict partial order. By the very definition of level sets, (G_f) is connected, each edge connects comparable vertices, and the planar embedding given by the actual level curves satisfies the face‑order condition. Hence any realizable pair must satisfy the two conditions.

For sufficiency, the authors assume a graph (G) and order (\le) that meet the two criteria. They first embed (G) in the plane without crossings, then triangulate each face (e.g., by adding diagonal edges) to obtain a simplicial complex. Next they assign scalar values (h(v)) to vertices consistent with the order: maximal vertices receive the highest values, minimal vertices the lowest, and intermediate vertices are placed by linear interpolation along comparable edges. On each triangle they define a piecewise‑linear function whose values at the vertices are those assigned. This yields a continuous, piecewise‑linear function (f_0) on the whole domain. To enforce the pseudoharmonic property, they apply a discrete Laplacian smoothing: for each interior vertex the value is replaced by the average of its neighbors, iterating until convergence. The resulting function (f) is continuous, has the prescribed ordering, and its level sets reproduce exactly the original graph structure because the smoothing does not alter the monotone ordering along edges and preserves the planar embedding of the contour network. Thus a pseudoharmonic function with invariant ((G,\le)) exists.

The paper also treats the case where the partial order is not total, allowing several vertices to share the same function value. In this situation the authors introduce a “multivalued” pseudoharmonic function, where a flat region corresponds to a face whose boundary vertices are mutually incomparable. They show that the same two conditions still guarantee realizability, extending the theory beyond strict total orders.

Several illustrative examples are provided. A simple tree reproduces known results. A graph containing a single cycle demonstrates that cycles are admissible provided the order respects the cyclic orientation (e.g., a clockwise monotone labeling). More complex graphs with multiple faces sharing edges are shown to be realizable when each face’s boundary respects the global order. These examples confirm that the theorem captures precisely the class of graphs that can arise from pseudoharmonic functions.

In the discussion, the authors point out potential applications. Since pseudoharmonic functions model physical potentials (electrostatic, fluid pressure, temperature) that are often sampled discretely, representing them by an invariant graph plus an order offers a compact, topologically faithful description. This can be exploited in image processing, mesh generation, and inverse problems where one wishes to reconstruct a continuous field from discrete measurements. Moreover, the constructive proof yields an algorithmic pipeline: embed the graph, triangulate, assign ordered vertex values, and perform Laplacian smoothing, thereby providing a practical method for synthesizing pseudoharmonic fields with prescribed contour topology.

The paper concludes by emphasizing that the two conditions are both necessary and sufficient, thereby completing the characterization. Future work is suggested in three directions: (i) extending the theory to higher‑dimensional domains where level surfaces become manifolds; (ii) incorporating nonlinear elliptic operators to model more general physical phenomena; and (iii) developing efficient computational implementations and error analyses for the reconstruction algorithm. Overall, the work bridges graph theory, planar topology, and potential theory, offering a rigorous foundation for the use of graphs as invariants of pseudoharmonic functions.