Questionnaire-2024/smt/2003_sol.tex

\begin{center}
  \begin{egraph}
    \node[base node] (a)                    {$a$};
    \node[base node] (b) [above right of=a] {$b$};
    \node[base node] (c) [below right of=b] {$c$};
    \node[base node] (d) [below of=c]       {$d$};
    \node[base node] (e) [below left of=d]  {$e$};
    \node[base node] (f) [left of=e]        {$f$};
    \node[base node] (g) [above left of=f]  {$g$};
    \path[]
        (a) edge [] node {} (b)
        (b) edge [] node {} (c)
        (c) edge [] node {} (d)
        (d) edge [] node {} (e)
        (e) edge [] node {} (a)
        (f) edge [] node {} (e)
        (g) edge [] node {} (f);

     \chord{b}{e};
     \chord{c}{e};
  \end{egraph}
\end{center}

\begin{align*}
  \varphi_{TC} =~\egraphFuncConstraint{a}{b}{e}~\land\\[0.75ex]
                 \egraphFuncConstraint{b}{c}{e}~\land\\[0.75ex]
                 \egraphFuncConstraint{c}{d}{e}
\end{align*}

\begin{align*}
  \hat\varphi_{E} \coloneqq
      \lnot e_{a=b} \land e_{b=c} \lor e_{c=d} \imp \lnot (\lnot e_{d=e} \lor e_{e=f}) \land \lnot(e_{f=g} \land \lnot e_{a=e})
\end{align*}
\begin{align*}
 \varphi_{prop} \coloneqq \varphi_{TC} \land \hat\varphi_{E}
\end{align*}