# Name for specific cycles in graphs

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 facets because vertices and edges that constitute to the boundary of a facet of a polyhedron are prototypical examples of such cycles.

• @MarkSapir the term convex subgraph exists but has a slightly different meaning: all shortest paths from $G$ must me in $C$. – M. Winter Mar 24 at 15:44

## 1 Answer

You are looking for the following:

Definition. A subgraph $$H\subseteq G$$ is called isometric if $$\mathrm{dist}_H(u,v)=\mathrm{dist}_G(u,v)$$ for all $$u,v\in V(H)$$.

So your cycles could be called isometric cycles.

Note that not all facets of a polyhedron are induced in this sense. Consider the a $$2n$$-gonal pyramid for some $$n\ge 3$$. Two antipodal vertices of the $$2n$$-gonal face $$C$$ have distance $$n$$ along $$C$$, but only distance two in the pyramid (via the apex vertex).