Is there an established name for cycles $C\subseteq G(V,E)$ with the property that

$$\lbrace u,v\rbrace\subseteq C\cap V\implies\mathrm{dist}_{|C}(u,v)\le \mathrm{dist}_{|G}(u,v)$$

I would be tempted to call them *facet*s because vertices and edges that constitute to the boundary of a facet of a polyhedron are prototypical examples of such cycles.

convex subgraphexists but has a slightly different meaning: all shortest paths from $G$ must me in $C$. $\endgroup$ – M. Winter Mar 24 at 15:44