Publication Type: Conference Paper/Unpublished Manuscript
Abstract: Recall that the distance of a vertex $x$ is $ d(x) = \sum_{\forall u
\in V(G)} d(x,u).$ We define the {\it $2$-stop-return} distance of
$x$ with two stops at $y$ and $z$ ($y \neq z$) to be
$$d_{2s} (x,\{y,z\}) = d(x,y) + d(y,z) + d(z,x).$$ %For simplicity, we will write $d_{2s}(x,y,z)$ instead of $d_{2s}(x,\{y,z\})$.

\vspace{.5cm} \noindent The {\it $2$-stop-return eccentricity
$e_{2s}(x)$} of a vertex $x$ in a graph $G$ is the maximum
$2$-stop-return distance from $x$, that is,
$$e_{2s} (x) = \max_{y, z \in V(G)} \bigg(d(x,y) + d(y,z) + d(z,x)\bigg).$$
The minimum $2$-stop-return eccentricity among the vertices of $G$
is the {\it $2$-stop-return radius} that is, $\rm rad_{2s} (G) =
\min_{x \in V(G)}e_{2s} (x)$. The maximum $2$-stop-return
eccentricity among the vertices of $G$ is the {\it $2$-stop-return
diameter} that is, $\rm diam_{2s} (G)$ $=$ $\max_{x \in V(G)} e_{2s}
(x).$




\vspace{.3cm}

\noindent This particular metric minimizes the distance that a pizza
delivery guy would need to travel if he wants to make two deliveries
in one trip. We present results about the $2$-stop-return distance
in graphs.