A characterization of graphs of radius-$r$ flip-width at most $2$

TL;DR

This paper characterizes graphs with radius-$r$ flip-width at most 2, showing they are exactly the ($C_5$, bull, gem, co-gem)-free graphs, which are totally decomposable by bi-joins, linking a new parameter to a well-studied graph class.

Contribution

It provides a precise characterization of graphs with flip-width at most 2 for all relevant radii, connecting a new graph parameter to known graph classes.

Findings

Graphs with $r$-flip-width ≤ 2 are exactly ($C_5$, bull, gem, co-gem)-free.

This class corresponds to totally decomposable graphs via bi-joins.

The result applies for all $r eq 1$ and $r= extinfty$.

Abstract

The $r$-flip-width of a graph, for $r\in \mathbb{N}\cup \{\infty\}$, is a graph parameter defined in terms of a variant of the cops and robber game, called the flipper game, and it was introduced by Toru\'{n}czyk (FOCS 2023). We prove that for every $r\in (\mathbb{N}\setminus \{1\})\cup \{\infty\}$, the class of graphs of $r$-flip-width at most $2$ is exactly the class of ($C_5$, bull, gem, co-gem)-free graphs, which are known as totally decomposable graphs with respect to bi-joins.