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

Abstract

A graph $G$ is unstable if its canonical double cover, CDC$(G)$, has strictly more automorphisms than Aut$(G)\times\mathbb{Z}_2$. A related question is whether two non-isomorphic graphs can share the same CDC. We place both problems in a unified framework of lifting and guided folding, showing that both are governed by conjugacy classes of strongly switching involutions in Aut(CDC$(G)$). Our approach uses two-fold isomorphisms (TF-isomorphisms), together with lifting and guided folding adapted from voltage-graph theory. Lifting a TF-isomorphism $(\alpha,\beta):G\to H$ produces a digraph isomorphic to the alternating double cover of $G$. Folding it back yields a graph TF-isomorphic to $G$: if the result is non-isomorphic to $G$, the two form a TF-cousin pair; if it coincides with $G$, then $(\alpha,\beta)$ is a non-trivial TF-automorphism and $G$ is unstable. Each guide corresponds to a switching involution of Aut(CDC$(G)$), and distinct conjugacy classes can produce distinct non-isomorphic base graphs sharing the same CDC. The framework generates TF-cousin pairs and unstable graphs from the seed pair $(C_k\cup C_k,C_{2k})$ for odd $k$. We introduce the claw graph family CG$(n)$ and prove that CG$(n)$ and its companion CG$'(n)$ are TF-cousins if and only if $n$ is odd. For $n=1$ the pair consists of the Petersen graph and a companion cubic graph on 10 vertices, with the Desargues graph as their common CDC. For each odd $n\geq3$ the construction yields a new pair of non-isomorphic cubic graphs sharing the same CDC. We conjecture that in every TF-cousin pair one member contains two vertex-disjoint copies of $C_k$ and the other contains $C_{2k}$ for some odd $k$, and that every unstable asymmetric graph contains both $C_k$ and $C_{2k}$ for some odd $k$. The first conjecture has been verified computationally for all connected graphs on at most 9 vertices.