A graph is intrinsically projectively linked (IPL) if its every embedding in projective space contains a nonsplit link. Some minor-minimal IPL graphs have been found previously. We determine that no minor-minimal IPL graphs on 16 edges exists and identify new minor-minimal IPL graphs by applying $\Delta-Y$ exchanges to $K_{ 7}-2e$. We prove that for a nonouter-projective-planar graph $G$, $G+\bar{ K}_{ 2}$ is IPL and describe the necessary and sufficient conditions on a projective planar graph $G$ such that $G+\bar{ K}_{ 2}$ is IPL. Lastly, we deduce conditions for $f(G + \bar{ K_{ 2}})$ to have no nonsplit link, where $G$ is projective planar, $\bar{ K_{ 2}} = \{ w_{ 0},w_{ 1}\}$, and $f(G + \bar{ K_{ 2}})$ is the embedding onto $\mathbb{ R}P^{ 3}$ with $f(G)$ in $z=0$, $w_{ 0}$ above $z=0$, and $w_{ 1}$ below $z=0$ such that every edge connecting ${ w_{ 0},w_{ 1}}$ to $G$ avoids the boundary of the 3-ball, whose antipodal points are identified to obtain projective space.
Categories: math.CO math.GT