The Excluded Vertex-Minors and Pivot-Minors for Rank-Width at Most Two

2026-07-25

AI Summary

This study completely characterizes the forbidden subgraph structures for the class of graphs of rank-width at most two. By iteratively applying the pivot-minor reduction theorem in reverse, the authors construct prime graph extensions layer by layer and systematically identify all forbidden vertex-minors (25 in total) and forbidden pivot-minors (609 in total), leveraging local equivalence and pivot equivalence relations. A computer-assisted proof framework is employed, utilizing isotropic systems to generate canonical labels for efficient equivalence-class classification. Exhaustive verification over more than $10^{11}$ candidate graphs confirms that no new forbidden vertex-minors exist on 11–16 vertices and no new forbidden pivot-minors arise on 13–16 vertices, thereby yielding—for the first time—a complete list of forbidden minors characterizing this graph class.

Abstract

We determine both the excluded vertex-minors and the excluded pivot-minors for the class of graphs of rank-width at most two. Up to local equivalence and graph isomorphism, there are exactly 25 excluded vertex-minors: 1 graph on 8 vertices, 18 on 9 vertices, and 6 on 10 vertices. Up to pivot equivalence and graph isomorphism, there are exactly 609 excluded pivot-minors: 2 on 8 vertices, 447 on 9 vertices, 146 on 10 vertices, 10 on 11 vertices, and 4 on 12 vertices. No excluded vertex-minor occurs on 11--16 vertices, and no excluded pivot-minor occurs on 13--16 vertices; the author's 16-vertex bound makes both lists complete. The proof is computer-assisted. Instead of enumerating all graphs, we reverse the one-vertex reduction theorem for prime graphs. For each $n$, we retain exactly the prime $n$-vertex graphs of rank-width at most two, modulo local equivalence and isomorphism, and extend those graphs by one vertex. Local-equivalence classes are identified by an exact canonical key obtained from the associated isotropic system, the binary row space of $[I\mid A(G)]$. A restricted version of the same key classifies pivot equivalence exactly. The vertex-minor and pivot-minor computations examine, respectively, more than $9.0\times 10^{10}$ and $4.9\times 10^{11}$ prime extensions in their 16-vertex final layers.

Problem

• rank-width
• vertex-minors
• pivot-minors
• excluded minors
• graph theory

Innovation

• rank-width
• vertex-minor
• pivot-minor
• isotropic system
• canonical key