I'm not sure that your question is very precise. You write "shortened" line because you use the option shorten? in fact you increase the line with shorten >=-1cm
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture} [dot/.style={draw,circle,minimum size=3pt,
inner sep=0pt,outer sep=0pt,thick,fill}}]
\coordinate[dot] (C) at (0,0);
\coordinate[dot] (CENTER) at (10,0);
\draw[style={shorten >=-1cm}, dashed] (C) -- (CENTER) ;
\end{tikzpicture}
\end{document}
gives
logically the simplest way
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\coordinate (C) at (0,0);
\node (CENTER) at (10,0) {NAME};
\draw[dashed] (C) -- (CENTER) ;
\end{tikzpicture}
\end{document}

But perhaps you really want to work against the effect of \draw[shorten >=-1cm, you can get a fine result with :
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[dot/.style={draw,circle,minimum size=3pt,
inner sep=0pt,outer sep=0pt,thick,fill}}]
\coordinate[dot] (C) at (0,0);
\coordinate[dot] (CENTER) at (10,0);
\draw[style={shorten >=-1cm}, dashed] (C) -- (CENTER) node[right=1cm] {NAME};
\end{tikzpicture}
\end{document}
This is one method to place a node relatively to another on. First when you write rightthe new node is placed with 'anchor=west` then if you decided to increase the line with 1 cm, you you need to move the new node to 1 cm to the right of CENTER.

\tikzset{add/.style args={#1 and #2}{to path={($(\tikztostart)!-#1!(\tikztotarget)$)--($(\tikztotarget)!-#2!(\tikztostart)$)\tikztonodes}}}and I discover with your example that I can use 1cm or 2cm. I think you can use the original style with0 and 1cm( I always use percent 0.1 or 1.2 etc. )if you want to increase the line for only one side. But the solution of Gonzalo and me are correct. – Alain Matthes Apr 23 '12 at 23:06