Planarity of Mycielski-like graph expansions

TL;DR

This paper characterizes graphs whose great shadow constructions are planar, revealing that such graphs are always bipartite cactus graphs, with implications for circuit routing in keyboard design.

Contribution

It provides a complete characterization of graphs with planar great shadows, linking planarity to bipartite cactus graph structure.

Findings

Graphs with planar great shadows are bipartite cactus graphs.

Planarity of great shadows determines circuit routing feasibility.

Characterization aids in designing printable circuit layouts.

Abstract

For a graph $G$, we define its great shadow $S(G)$ as a construction that duplicates each vertex $v$ in $G$ and sets this duplicated vertex adjacent to $v$ and all neighbors of $v$. Great graph shadows arise naturally in the routing of diode-and-switch circuits for computer keyboards, and are closely related to the Mycielski operation. These diode-and-switch circuits can be routed on a single-sided printed-circuit board if and only if the corresponding great shadow is planar. In this paper, we characterize all graphs with planar great shadows. Such graphs are always bipartite cactus graphs.