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Pizza Delivery: 2-Stop-Return Distances in Graphs

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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.
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Association:
Name: The Mathematical Association of America MathFest
URL:
http://www.maa.org


Citation:
URL: http://www.allacademic.com/meta/p275715_index.html
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MLA Citation:

Gera, Ralucca., Eroh, Linda., Winters, Steven. and Bullington, Grady. "Pizza Delivery: 2-Stop-Return Distances in Graphs" Paper presented at the annual meeting of the The Mathematical Association of America MathFest, TBA, Madison, Wisconsin, Jul 28, 2008 <Not Available>. 2009-05-23 <http://www.allacademic.com/meta/p275715_index.html>

APA Citation:

Gera, R. , Eroh, L. , Winters, S. J. and Bullington, G. , 2008-07-28 "Pizza Delivery: 2-Stop-Return Distances in Graphs" Paper presented at the annual meeting of the The Mathematical Association of America MathFest, TBA, Madison, Wisconsin <Not Available>. 2009-05-23 from http://www.allacademic.com/meta/p275715_index.html

Publication Type: Conference Paper/Unpublished Manuscript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.

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