Deformations of dotted graphs consisting of standard circles

TL;DR

This paper studies how certain finite graphs with degree-2 vertices, called dotted graphs, can be deformed or reduced, focusing on those associated with lattice polytopes and composed of standard circles.

Contribution

It introduces a framework for analyzing the reducibility of admissible dotted graphs with standard circles, expanding understanding of their deformation properties.

Findings

Characterization of reducibility conditions for dotted graphs

Identification of deformation types applicable to standard circle configurations

Insights into the structure of lattice polytope-associated graphs

Abstract

Dotted graphs are certain finite graphs with vertices of degree 2 called dots in the $xy$-plane $\mathbb{R}^2$, and a dotted graph is said to be admissible if it is associated with a lattice polytope in $\mathbb{R}^2$ each of whose edge is parallel to the $x$-axis or the $y$-axis. A dotted graph is said to be reducible if certain types of deformations are applicable. In this paper, we investigate the reducibility of admissible dotted graphs in certain simple forms consisting of standard circles.