  (Abridged) Binary systems with similar components are ideal laboratories
which allow several physical processes to be tested, such as the possible
chemical pattern imprinted by the planet formation process. Aims. We explore
the probable chemical signature of planet formation in the remarkable binary
system HD 80606 - HD 80607. The star HD 80606 hosts a giant planet with 4 MJup
detected by both transit and radial velocity techniques, being one of the most
eccentric planets detected to date. We study condensation temperature Tc trends
of volatile and refractory element abundances to determine whether there is a
depletion of refractories that could be related to the terrestrial planet
formation. Methods. We carried out a high-precision abundance determination in
both components of the binary system, using a line-by-line strictly
differential approach, using the Sun as a reference and then using HD 80606 as
reference. We used an updated version of the program FUNDPAR, together with
ATLAS9 model atmospheres and the MOOG code. Conclusions. From the study of Tc
trends, we concluded that the stars HD 80606 and HD 80607 do not seem to be
depleted in refractory elements, which is different for the case of the Sun.
Then, the terrestrial planet formation would have been less efficient in the
components of this binary system than in the Sun. The lack of a trend for
refractory elements with Tc between both stars implies that the presence of a
giant planet do not neccesarily imprint a chemical signature in their host
stars, similar to the recent result of Liu et al. (2014). This is also in
agreement with Melendez et al. (2009), who suggest that the presence of
close-in giant planets might prevent the formation of terrestrial planets.
Finally, we speculate about a possible planet around the star HD 80607.
