با سلام . این کار با دو دستور\stepwise و \step انجام پذیر است
\tikzstyle{vertex}=[circle, draw, inner sep=0pt, minimum size=22pt]
\newcommand{\vertex}{\node[vertex]}
\tikzstyle{vertexs}=[draw,minimum width=.6cm,minimum height=.6cm]
\newcommand{\vertexs}{\node[vertexs]}
\begin{center}
\stepwise
{%
\begin{tikzpicture}
\step{
\vertexs (a1) at (0,0) [label=above:$c_{1}$]{};
\vertexs (a2) at (1.3,0) [label=above:$c_{2}$]{};
\vertexs (a3) at (2.6,0) [label=above:$c_{3}$]{};
\vertexs (a4) at (3.9,0) [label=above:$c_{4}$]{};
\vertex (b1) at (-1.3,-3) [label=below:$v_{1}$]{};
\vertex (b2) at (0,-3) [label=below:$v_{2}$]{};
\vertex(b3) at (1.3,-3) [label=below:$v_{3}$]{};
\vertex (b4) at (2.6,-3) [label=below:$v_{4}$]{};
\vertex (b5) at (3.9,-3) [label=below:$v_{5}$]{};
\vertex (b6) at (5.2,-3) [label=below:$v_{6}$]{};
\path
(a1) edge [gray] (b1)
(a1) edge [gray] (b2)
(a1) edge [gray] (b4)
(a2) edge [gray] (b2)
(a2) edge (b3)
(a2) edge (b5)
(a3) edge [gray] (b1)
(a3) edge (b5)
(a3) edge (b6)
(a4) edge (b3)
(a4) edge [gray] (b4)
(a4) edge (b6);}
\step{\vertexs [fill=green](a2) at (1.3,0) [label=above:$c_{2}$]{};}
\step{ \path
(a2) edge [ultra thick,green](b3)
(a2) edge [ultra thick,green](b5);}
\end{tikzpicture}
}
\end{center}