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

Abstract

The radius-r flip-width of a graph, for $r\in N\cup \{\infty \}$, 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 Toruńczyk (FOCS 2023). We prove that for every $r\in (N\setminus \{1\left\}\right)\cup \{\infty \}$, the class of graphs of radius-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.