One-Vertex Lifts of Looped Circle Graphs: Obstructions and Interlace Polynomials
Let $M$ be the adjacency matrix of a looped simple graph over $\mathbb{F}_2$ and let $t=\operatorname{diag} M$. We study the one-vertex lift with adjacency matrix $\left(\begin{smallmatrix}M+tt^T&t\\t^T&0\end{smallmatrix}\right)$. A looped circle graph is called stable if its lift is a circle graph. We prove that stability is equivalent to the existence of a pseudo-orientable bouquet realizing the given looped circle graph. We then obtain a finite excluded toggle-minor characterization by applying two rooted constructions to the known excluded pivot-minors for circle graphs. An exact enumeration yields $22$ minimal toggle-equivalence classes: one of order $5$, twelve of order $6$, and nine of order $7$. The enumeration is accompanied by independently checkable linear-algebra and minor certificates. Finally, we prove $q(G^\dagger;y)=yq(G;y)$ for the vertex-nullity interlace polynomial of every looped simple graph, and derive quasi-tree counting and real determinant formulas.