Effect of graph operations on graph associahedra

TL;DR

This paper investigates how specific graph operations influence the structure of rotation graphs of graphs, providing new bounds, exact diameters for certain classes, and insights into chromatic numbers for various graph families.

Contribution

It characterizes the effects of adding vertices and twins on rotation graphs and derives new bounds and exact values for diameters and chromatic numbers.

Findings

Chromatic number of rotation graphs of threshold and complete bipartite graphs is 3.

Established conditions preserving chromatic number under graph operations.

Computed exact diameters for rotation graphs of certain complete bipartite graphs.

Abstract

Given a graph $G$, we determine the structure of the rotation graph of a graph obtained by applying certain operations to $G$. Specifically, we consider the operations of adding a simplicial vertex, adding a true twin to a vertex, and the two closely related operations of deleting the set of edges from a subgraph induced by a set of true twins, and adding a false twin to a vertex. We describe how applying these operations to a graph affects the structure of its rotation graph. Furthermore, by using this description, we study chromatic number, distance, and diameter in rotation graphs. In particular, we establish conditions under which the chromatic number of the rotation graphs is preserved. As an interesting consequence, we obtain that the chromatic number of the rotation graphs of threshold graphs (which includes complete split graphs and star graphs) and of complete bipartite graphs is 3. We also provide a new lower bound for $\text{diam}(\mathcal{R}(G-S))$ in terms of $\text{diam}(\mathcal{R}(G))$, where $S$ is the set of edges of the subgraph of $G$ induced by a set of true twins. As a consequence, we improve the known lower bound for the diameter of the rotation graph of balanced complete bipartite graphs, allowing us to compute the exact value of $\text{diam}(\mathcal{R}(K_{2,q}))$ for $q\in\{3,4,5,6,7,8\}$.