2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159857We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in combinatorial and knot theoretic languages.30 pages, many figuresCombinatoricsGeometric Topology05C10; 57M25text