Seymour's self-minor conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For graphs $H$ and $G$, a minor model consists of pairwise disjoint nonempty connected vertex sets $B_v\subseteq V(G)$, one for each $v\in V(H)$, with an edge between $B_u$ and $B_v$ whenever $uv\in E(H)$. A proper minor model does not merely reproduce the whole graph with singleton branch sets and every edge: it deletes a vertex or edge, or contracts a nontrivial connected branch set. Thus a proper self-minor is a proper minor model of a graph isomorphic to $G$. This does not require the resulting graph to be nonisomorphic to $G$.

Statement recorded in the supplied source.

Does every infinite graph have a proper minor isomorphic to itself? The source also isolates the countably infinite case of the same question after reporting that Oporowski's counterexample to the unrestricted assertion is uncountable. [S2]

Short English statement

Can every infinite graph be obtained, up to isomorphism, by a nontrivial minor operation on itself, and does this at least hold for countably infinite graphs?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3297 (OPG-333). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Seymour's self-minor conjecture, node 333, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/seymours_self_minor_conjecture.