TikZ: shaded area enveloping a graph - graph

Question

I need to draw a shaded area enveloping a graph like in the example below. I am looking for an automated procedure and a more satisfying result. Compared to the example, I need smoother edges and more regular distances between vertices (circles) and the boundaries of the area.
Drawing arbitrary shapes in TikZ are time consuming, I suspect that a smart trick might do the job, like creating large overlapping areas centered at each vertex. Any suggestions?
And here is a minimal working LaTeX code to generate the graph which has to be enveloped:
\documentclass[tikz]{standalone}
\usepackage{tikz}
\begin{document}
\tikzstyle{vertex}=[circle,draw=black,thick]
\begin{tikzpicture}
\foreach \pos/\name in {{(0,1)/1}, {(1,1)/2}, {(0,0)/3}, {(1,0)/4}, {(2,0)/5}, {(1,-.7)/6}}
\node[vertex] (\name) at \pos {};
\foreach \one/\two in {{1/2}, {1/3}, {1/4}, {1/6}, {2/3}, {2/4}, {2/5}, {3/6}, {4/6}, {4/5}}
\draw[thick] (\one) -- (\two);
\end{tikzpicture}
\end{document}

Answer 1

You could connect your nodes with a polygon with a tick stroke:
\documentclass[border=2mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{backgrounds}
\begin{document}
\tikzset{vertex/.style={circle,draw=black,thick}}
\begin{tikzpicture}
\foreach \pos/\name in {{(0,1)/1}, {(1,1)/2}, {(0,0)/3}, {(1,0)/4}, {(2,0)/5}, {(1,-.7)/6}}
\node[vertex] (\name) at \pos {};
\foreach \one/\two in {{1/2}, {1/3}, {1/4}, {1/6}, {2/3}, {2/4}, {2/5}, {3/6}, {4/6}, {4/5}}
\draw[thick] (\one) -- (\two);
\begin{scope}[on background layer]
\filldraw[lightgray,line width=20pt,rounded corners=5pt] (1.center) -- (2.center) -- (5.center) -- (4.center) -- (6.center) -- (3.center) -- cycle;
\end{scope}
\end{tikzpicture}
\end{document}