As a rule, for identifying groups by different selection methods, the principle of overdensity in comparison with the background is used. The richer the group population, the stronger the overdensity and therefore the more likely that such a group is physically bound. Identification of poor groups depends strongly on the limiting parameters of the method.
Melnyk, Elyiv & Vavilova (2006) demonstrated that the first-order tessellation is more useful for searching the rich clusters of galaxies than the small groups. In the first-order Voronoi tessellation, the key parameter is the volume of the galaxy’s Voronoi cell V. This parameter characterizes a galaxy environmental density. The condition of cluster/group membership of a certain galaxy is the
relatively small value of V. This condition is true when the galaxy is surrounded by close neighbouring galaxies. That is why the first order Voronoi tessellation is not corrected for the identification of small isolated galaxy systems. In our paper Elyiv, Melnyk & Vavilova (2009) “High-order 3D Voronoi tessellation for identifying isolated galaxies, pairs and triplets”, we propose the high-order 3D Voronoi tessellation method for identification of pairs and triplets.
Contrary to the first-order tessellation (Fig. 3a), the second-order tessellation for set S distribution of nuclei is the partition of the space which associates a region V1,2 with each pair of nuclei 1 and 2 from S in such a way that all points in V1,2 are closer to 1 and 2 than other nuclei from S (Fig. 3b). Region V1,2 is a common cell for nuclei 1 and 2. In such a way, the second-order Voronoi tessellation is available for the identification of single galaxies and pairs. The third-order Voronoi tessellation is appropriate for the identification of triplets. It is the partition of the space which associates a region V1,2,3 with each triplet of nuclei 1, 2, 3 in such a way that all points in V1,2,3 are closer to nuclei 1, 2, 3 than other nuclei from S (Lindenbergh 2002), Fig. 3c.
Therefore we applied high-order 3D Voronoi tessellation to the sample of galaxies from the SDSS SR 5 spectroscopic survey 2500 to 10000 km/s (see all details in Elyiv et al. 2009). This approach allows us to select small galaxy groups and isolated galaxies in different environments and to find the isolated systems. We also considered some physical properties of selected galaxy systems and concluded that in such small groups as pairs and triplets, segregation by luminosity is clearly observed: galaxies in isolated pairs and triplets are on average two times more luminous than isolated galaxies. We also found that systems in denser environments have greater rms velocity and