In this paper chain cacti are considered. First, for two specific classes of chain cacti (orto-chains and meta-chains of cycles with h vertices) the recurrence relation for independence polynomial is derived. That recurrence relation is then used in deriving explicit expressions for independence number and number of maximum independent sets for such chains. Also, the recurrence relation for total number of independent sets for such graphs is derived. Finaly, the proof is provided that orto-chains and meta-chains are the only extremal chain cacti with respect to total number of independent sets (orto-chains minimal and meta-chains maximal).

Background. Vitiligo is a common skin disorder characterized by macular depigmentation of the skin. The etiopathogenesis of the disease is still unclear, but there is evidence that autoimmunity and endocrine disfunction may be involved. Objective. The aim of this study was to determine whether vitiligo is statistically associated with thyroid autoimmunity. Method. In a prospective case-control study, we compared the frequency of thyroid autoantibodies (thyroglobulin antibody, anti-Tg and thyroid peroxidase antibody, and anti-TPO) in 33 patients with vitiligo and in 33 healthy volunteers. Thyroid autoantibodies and thyroid hormones (thyroxine (T4), triiodothyronine (T3), and thyroid stimulating hormone (TSH) were measured in all subjects. Results. Thyroid functional abnormalities were found in 6 (18.18%) patients. Anti-Tg and anti-TPO were positive in 9 (27.27%) and 8 (24.24%) patients, respectively. In control group, only one subject (3.03%) had abnormalities in thyroid hormonal status, and two subjects had positive thyroid autoantibodies. Compared with the control group, the frequency of both anti-Tg and anti-TPO was significantly higher in those with vitiligo (P < .05). Conclusion. This study shows a significant association between vitiligo and thyroid autoimmunity, and that tests to detect thyroid autoantibodies are relevant in patients with vitiligo.