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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.

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up vote 1 down vote accepted

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 $y.$

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