tokens: from four Gaussian functions in the surface brightness − colour plane. Notes. The MP values quoted in the second column are the mixing proportions for each component of the mixture model. For the mean vectors and covariance matrices, the x variable corresponds to the galaxy colour g − r , while the y variable is the surface brightness μ r . Table C.2. Galaxies in the SDSS DR14 photometric catalogue around compact groups that are potential sources of contamination. We measured the surface brightnesses ( μ r ) and observer-frame colours ( g − r ) of each spectroscopic galaxy and of the potentially contaminating galaxies. The surface brightness of each galaxy was computed using the Petrosian radius provided by the SDSS pipeline. After analysing the surface brightness − colour relation, we used the Mclust package ( Scrucca et al. 2017 ) of R software to perform a density estimation based on finite Gaussian mixture modelling. From its application to our data in the surface brightness − colour plane, we observed that, according to the Bayesian Information Criterion, a proper density representation can be obtained using a mixture of four Gaussian functions for a model with ellipsoidal distribution that allows variable volume, shape, and orientation. The best fitted parameters for the obtained Gaussian mixture modelling are quoted in Table C.1 , while the perspective plot of the resulting surface density is shown in the upper panel of Fig. C.1 . Using this surface density, we defined the isodensity contour that enclosed 95% of the spectroscopic galaxies in the surface brightness − colour plane. This region is represented as the grey. shaded area in the lower panel of Fig. C.1 . We also show the scatter plot for the potential sources of contamination (red dots) Photometric galaxies lying outside the grey region are considered non-contaminating galaxies (empty red dots). This criterion allowed us to discard 104 galaxies. Finally, as described in Sect. 4.2 , after a visual inspection of the remaining 81 objects, we confirm that 13 were actually objects misclassified as galaxies by the SDSS pipeline. Therefore, only 68 galaxies are kept as potential sources of contamination. The SDSS object identification numbers of these 68 galaxies are quoted in Table C.2 . Appendix D: New CG catalogue in the SDSS In this appendix we present the tables of HMCGs identified in the SDSS. In Table D.1 we show the content for the table containing 462 HMCGs, while Table D.2 shows the content for the table that contains the information for the 2070 galaxy members. Table D.1. Compact groups identified in the SDSS DR12 using the modified algorithm. Notes. HMCGid: group ID, N : number of galaxy members, RA: group centre right ascension (J2000), Dec: group centre declination (J2000), Redshift: group CMB redshift, Θ G : angular diameter of the smallest circumscribed circle, μ r : r -band group surface brightness, σ v : radial velocity dispersion, r b : r -band observer-frame
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