The S-rule and 1-d representation for the traversal of a planar graph in AEC industry

TL;DR

This paper introduces the S-rule traversal method and a 1-dimensional graph expression for planar graphs, aligning with industry cognition, enabling efficient visualization, combination, and advanced calculations like FEM.

Contribution

It proposes a novel traversal and representation method for planar graphs that preserves topological relationships and facilitates industry-specific visualization and computation.

Findings

The 1-d expression retains the topological structure of planar graphs.

The method supports conversion between 1-d and 2-d forms for visualization.

Applicable to higher-dimensional graphs by reduction to 1 or 2 dimensions.

Abstract

Based on two trivial observations of the AEC industry, this paper proposes a traversal method ("S-rule") and expression ("1-dimensional-graph") transformed from DFS. This traversal method conforms to the original cognitive logic of the AEC industry, and the 1-d expression has clear language characteristics while completely retaining the topological relationship of the planar graph : a sequence of finite symbols (vocabularies) under definite rules. Moreover, the language can be restored to a standard 2-d form that is isomorphic to the original planar graph, thus ensuring its visualization characteristics. Fragments of the 1-d language can be used as planar units for free combination and weighting, and as the data foundation to support advanced calculations including FEM and isomorphic matching. And after the 2-d graph is reduced to 1-d, any 3-d or higher-dimensional graphs can also be reduced to 1 or 2 dimensions. The first half of this paper (Chapter 1) takes the 4X4 standard grid as an example to introduce the prototype of S-rule and 1-d expression, and gives the mapping rule from 1-d expression to its editable text form. In the second half of this paper, the rule and expression are gradually extended to non-embedded planar graphs (Chapter 2) and embedded planar graphs (Chapter 3), and the "grammar" is finally summarized (Chapter 4).