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It is know that the Gabriel graph of a point set $P \subset \mathbb{R}^2$, $\mathcal{GG}(P)$ is a subset of the corresponding Delauney graph $\mathcal{DG}(P)$, i.e. $\mathcal{GG}(P) \subseteq\mathcal{DG}(P)$.

A Gabriel graph is defined as:

The Gabriel graph $\mathcal{GG}(P)$ of a point set $P \subset \mathbb{R}^2$ is defined as follows. The vertex set is $P$ and two vertices $p, q$ in $P$ are connected by an edge if and only if the interior of the disk with diameter $\overline{pq}$ is empty and $p, q$ are the only two points on its boundary. In particular, this implies that $\overline{pq}$ is also an edge in the Delaunay graph of P.

However, I am struggling to visualize or understand in what cases an edge $(p, q)$ would be in $\mathcal{DG}(P)$ but not in $\mathcal{GG}(P)$. Could someone give me an example? It seems to me as, by the empty-circle property of Voronoi edges, and their duality with the Delaunay graph, every edge in the Delaunay graph should satisfy the definition of a Gabriel graph. It is obvious that I am missing some edge case.

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    $\begingroup$ Note that the smallest disk containing a triangle and the smallest disk containing an edge of this triangle are, in general, not the same. Draw a triangle, all the disks for the Gabriel graph and the disk for the Delaunay graph. Can you place a point that does not destroy the Delaunay edge, but does destroy the Gabriel edge? $\endgroup$
    – Discrete lizard
    Dec 27 '21 at 13:05
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From my answer to a similar question:

A Delaunay edge $(x,y)$ won't be a Gabriel edge, if the set of all the possible empty disks with $x$ and $y$ on their boundaries doesn't contain the disk with minimum possible radius (which is the half-length of this edge). In other words, this will happen when the Delaunay edge $(x,y)$ doesn't intersect the segment (or ray), separating sites $x$ and $y$ in the Voronoi diagram.

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Ah, I think I figured it out, thanks to the help from Discrete Lizard.

See the following image of a Delaunay triangulation, where I've highlighted one example where the edges would be included in DT, but not GG.

A Delauney triangulation where one of the points is highlighted, as it would not be included in a Gabriel graph.

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