2 The CG identification algorithm

In order to safely deal with CG multiplicity and properly compare T and M characteristics we have devised a CG identification algorithm imposing compactness as the only requirement. The algorithm counts neighbours to each galaxy in 3D space within a volume defined by projected distance $\Delta r$ and velocity "distance'' $\Delta v^{\rm I} (\Delta r$ and $\Delta v^{\rm I}$ are free input parameters). When a galaxy is found to have at least two neighbours the geometrical center of the system is identified. Additional members within $\Delta r$ and $\Delta v^{\rm I}$ of the centroid are then searched for and a new center computed. This is an iterative process that goes on until convergence is reached, i.e. no further CG member is detected and no previously identified CG member gets excluded. Non-convergent systems are obviously rejected. CG centers are not weighted by magnitude of member galaxies on purpose, in order to enable non-biased investigation of possible relationships linking CG kinematics to luminosity.

The searching method is asymmetric and may produce different grouping depending on which galaxy is selected first. In order to overcome this undesirable effect the algorithm retains in the main sample only CGs whose single galaxies all have no further neighbour (within $\Delta r$ and $\Delta v^{\rm I}$) except those already listed as members. Non-symmetric CGs are excluded, because without definition of further selection criteria, the algorithm is unable to define which galaxies are CG members and which are to be left outside. The symmetrization procedure also ensures that no overlapping CGs are retained. Finally, cross correlation with ACO clusters (Struble & Rood 1999) enables the algorithm to exclude from the sample CGs which are cluster substructures at distance less than 1  $R_{\rm Abell}$ from the ACO centers.

For each CG the local surrounding galaxy density is computed within the free input parameters $\Delta R$ and $\Delta v^{\rm II}$. The algorithm also provides parameters indicative of average compactness and maximum physical extensions. These are the unbiased line of sight velocity dispersion $\sigma _{v}$, the maximum difference in redshift space between a CG member and the center $\Delta v_{\rm max}$, the radius ${\it r}_{\rm ave}$ measuring projected average galaxy distance from the center, and the radius ${\it r}_{\rm max}$ defined as the projected separation between the center and the most distant CG member galaxy. Average projected dimension of CGs ( $r_{\rm ave}$) is preferred to the median value, because having imposed a maximum physical extension to CGs, each galaxy distance should be equally weighted.

Our algorithm displays some analogies and differences with the friends of friends (FoF) group searching algorithm by Huchra & Geller (1982) and with the hierarchical procedure applied by Tully (1987). Like Tully (1987) our CGs are defined by internal conditions only and our procedure starts hierarchically by requiring a minimum galaxy density threshold to identify a CG. At variance with the FoF method, requiring a maximum galaxy-galaxy separation as a function of redshift, we impose a maximum size for the CGs. Adopting a common scale for structures allows to safely deal with multiplicity but induces a redshift luminosity dependence. To correct for this bias the CG sample is divided in 4 distance classes (see Sect. 3) and the comparison of CGs of different multiplicity is performed within each class. Moreover while the FoF procedure, to discriminate between physical and non physical systems, requires a minimum density contrast threshold (computed with respect to the average galaxy density of the sample), our CGs are identified without a constraint on density contrast. Instead, we do compute the surface density contrast locally (within $\Delta R$ and $\Delta v^{\rm II}$) after CGs have been identified. The advantage of this approach is that we can perform non biased analysis of CG environments.