% Math
\renewcommand{\vec}[1]{\bm{#1}}
\newcommand{\mathtuple}[1]{\left<#1\right>}
\DeclareMathOperator\cost{cost}
\DeclareMathOperator\edgeCost{c}
\DeclareMathOperator\minimum{min}
\DeclareMathOperator\cheapest{billigst}
\DeclareMathOperator\nodeDegree{deg}

% Some colors
\definecolor{darkpastelgreen}{rgb}{0.01, 0.75, 0.24}




% TikZ
\tikzstyle{thickArrow}=[line width=1mm, draw=black,-triangle 45,postaction={draw, line width=3mm, shorten >=4mm, -}]
% Allowes to draw a line parallel to an edge
% Example:
% 1. Declare a style in your graph: parallel/.style={decoration={sl,raise=-.75mm},decorate}
% 2. Use the style: \draw[-,parallel] (a) to (b);
%    This draws a line parallel to the edge a--b
\pgfdeclaredecoration{sl}{initial}{
	\state{initial}[width=\pgfdecoratedpathlength-1sp]{
		\pgfmoveto{\pgfpointorigin}
	}
	\state{final}{
		\pgflineto{\pgfpointorigin}
	}
}



% Beamer
\newcommand{\pageItemNote}[2]{
	\only<#1>{
		\note{
			\begin{itemize}
				#2
			\end{itemize}
		}
	}
}
\newcommand{\pageNote}[2]{
	\only<#1>{
		\note{
			#2
		}
	}
}