Lifting and Folding: A Framework for Unstable Graphs and TF-Cousins

Abstract

A graph $G$ is \emph{unstable} if its canonical double cover CDC$(G)$ has more automorphisms than Aut$(G)\times \mathbb{Z}_2$. A related problem asks when two non-isomorphic graphs share the same CDC. We unify both via \emph{lifting} and \emph{guided folding}, showing that they are governed by conjugacy classes of strongly switching involutions in Aut(\CDC$(G)$). Using \emph{two-fold isomorphisms} (TF-isomorphisms), lifting $(\alpha,\beta):G\to H$ produces a digraph isomorphic to the alternating double cover of $G$, while folding yields a graph TF-isomorphic to $G$. If this graph is non-isomorphic to $G$, the pair forms TF-cousins; otherwise $(\alpha,\beta)$ is a non-trivial TF-automorphism and $G$ is unstable. Distinct conjugacy classes of switching involutions in Aut$(CDC(G))$ produce non-isomorphic graphs with a common CDC, recovering a theorem of Pacco and Scapellato. The framework generates TF-cousin pairs and unstable graphs of arbitrary order from $(C_k\cup C_k,\, C_{2k})$. We introduce the \emph{claw graph} family CG$(n)$ and show that CG$(n)$ and CG'$(n)$ are TF-cousins iff $n$ is odd. For $n=1$, this yields the Petersen graph and a cubic companion on $10$ vertices, both with the Desargues graph as CDC. For odd $n\geq 3$, we obtain new non-isomorphic cubic graphs sharing a CDC. We conjecture that every TF-cousin pair and unstable graph contains cycles $C_k$ and $C_{2k}$ for some odd $k$, verified for all connected graphs on at most $9$ vertices.