For any circle graph H, every graph either has a vertex-minor isomorphic to k copies of H, or is a small perturbation of a graph with no H vertex-minor.
Beineke and Allen J
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The Erd\H{o}s-P\'{o}sa property for circle graphs as vertex-minors
For any circle graph H, every graph either has a vertex-minor isomorphic to k copies of H, or is a small perturbation of a graph with no H vertex-minor.