The Kelmans-Seymour conjecture III: 3-vertices in K4-

Let G be a 5-connected nonplanar graph and let x(1), x(2), y(1), y(2) is an element of V(G) be distinct, such that G[{x(1), x(2), y(1), y(2)}] congruent to K-4(-) and y(1)y(2) is not an element of (G). We show that one of the following holds: G - x(1) contains K-4(-), or G contains a K-4(-) in which x(1) is of degree 2, or G contains a TK5 in which x(1) is not a branch vertex, or {x(2), y(1), y(2)} may be chosen so that for any distinct z(0), z(1) is an element of N(x(1)) - {x(2),y(1),y(2)}, G - {x(1)v : v is not an element of{z(0),z(1),x(2),y(2),y(1)}} contains TK5. (C) 2019 Elsevier Inc. All rights reserved.