An <a href="https://en.wikipedia.org/wiki/Apex_graph" class="extiw" title="en:Apex graph">apex graph</a>, used by <a href="https://en.wikipedia.org/wiki/Neil_Robertson_(mathematician)" class="extiw" title="en:Neil Robertson (mathematician)">Neil Robertson</a> as an example of a graph that is <a href="https://en.wikipedia.org/wiki/linkless_embedding" class="extiw" title="en:linkless embedding">linklessly embeddable</a> but not YΔY-reducible. It can be formed by connecting a new vertex (the apex, shown in red) to each degree-three vertex of a <a href="https://en.wikipedia.org/wiki/Rhombic_dodecahedron" class="extiw" title="en:Rhombic dodecahedron">rhombic dodecahedron</a>, or by merging two opposite vertices of a four-dimensional <a href="https://en.wikipedia.org/wiki/Hypercube_graph" class="extiw" title="en:Hypercube graph">hypercube graph</a> to form a single supervertex. It is <a href="https://en.wikipedia.org/wiki/Triangle-free_graph" class="extiw" title="en:Triangle-free graph">triangle-free</a> and has minimum degree four, preventing any YΔY-reduction from applying to it.
<a href="//commons.wikimedia.org/wiki/User:David_Eppstein" class="mw-redirect" title="User:David Eppstein">David Eppstein</a>
<span class="int-own-work" lang="en">Own work</span>