Consider the following definition.


Let $C$ be a cycle of a simple graph $G$. We say that $C$ is *convex* if for any pair of distinct vertices $u,v \in V(C)$ $$ d\_C(u,v) < d\_{G-C}(u,v).$$


Is there any other name for such cycles? I was trying to find out some references/literature presenting results related to such cycles but I haven't found anything useful. I am mostly interested in the questions of whether such cycles have any other characterization and what is the structure of graphs that have many such cycles.



Definition of convex cycles
In the paper entitled Convex cycle bases and Cartesian products
 by Hellmuth, Leydold and Stadler, I found the following characterization of convex cycles 


Let $G$ be a simple graph and $C \subseteq G$ a cycle. If $|C|$ is odd then $C$ is convex if and only if for every edge $e = xy \in C$ there exist a unique vertex $z \in C$ such that 


$$ d\_G(x,z) = d\_G(y,z) = \frac{|C|-1}{2} \hbox{ and } S\_{xz} = S\_{yz} = 1.$$


If $|C|$ is even then $C$ is convex if and only if for every edge $e = xy \in C$ there is a unique edge $f = uv \in C$ such that


1. $ d\_G(x,u) = d\_G(y,v) = \frac{|C|}{2}-1 $
2. $d\_G(x,v) = d\_G(y,u) = \frac{|C|}{2}$
3. $ S\_{xu} = S\_{yv} = 1$
4. $ S\_{xv} = S\_{yu} = 2$


Where $S\_{xy}$ denotes the number of shortest paths between $x$ and $y.$


