Embedding Half-Edge Graphs in Punctured Surfaces

TL;DR

This paper extends the theory of graph embeddings to half-edge graphs in punctured surfaces, establishing conditions for their equivalence with half-edge ribbon graphs and exploring their duals, with implications for physics and mathematics.

Contribution

It generalizes the concept of graph embeddings to half-edge graphs in punctured surfaces and characterizes their equivalence with half-edge ribbon graphs.

Findings

Established conditions for embedding equivalence between half-edge graphs and ribbon graphs.

Identified the geometric duals of cellularly embedded half-edge graphs.

Extended the theory of graph embeddings to broader classes relevant in physics.

Abstract

It is known that graphs cellularly embedded into surfaces are equivalent to ribbon graphs. In this work, we generalize this statement to broader classes of graphs and surfaces. Half-edge graphs extend abstract graphs and are useful in quantum field theory in physics. On the other hand, ribbon graphs with half-edges generalize ribbon graphs and appear in a different type of field theory emanating from matrix models. We then give a sense of embeddings of half-edge graphs in punctured surfaces and determine (minimal/maximal) conditions for an equivalence between these embeddings and half-edge ribbon graphs. Given some assumptions on the embedding, the geometric dual of a cellularly embedded half-edge graph is also identified.