One-Vertex Lifts of Looped Circle Graphs: Obstructions and Interlace Polynomials

Let M be the adjacency matrix of a looped simple graph over F₂ and let t=diag M. We study the one-vertex lift with adjacency matrix (smallmatrixM+ttT&t&0smallmatrix). 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^;y)=yq(G;y) for the vertex-nullity interlace polynomial of every looped simple graph, and derive quasi-tree counting and real determinant formulas.