Eulerian and Hamiltonian Graphs · Hard problem

A graph consists of two separate pieces: a triangle on $\{1,2,3\}$, and a star with center $4$ and leaves $5,6,7$ (edges $45,46,47$) — the two pieces share no vertex and no edge. What is the minimum number of new edges (between any two of the $7$ vertices) needed so the WHOLE resulting graph is connected and has an Euler circuit?