Rhombic embeddings of planar graphs with faces of degree 4

TL;DR

This paper investigates conditions for embedding planar graphs with quadrilateral faces as rhombuses in the plane, providing criteria for existence and describing the set of all such embeddings.

Contribution

It establishes a necessary and sufficient condition for rhombic embeddings of planar graphs with degree 4 faces and characterizes all possible embeddings.

Findings

Derived a criterion for the existence of rhombic embeddings.

Characterized the entire set of such embeddings.

Provided a comprehensive description of embedding configurations.

Abstract

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings.