As a rule, there is no prohibition of yichud (seclusion) with one's own children. However, what about an adopted child? Since the parents and children are not actually related by blood, is there is problem for a father and his adopted daughter to be alone together in a house? We will review Rav Moshe Feinstein's answer today, and follow soon with the more detailed analysis of the Tzitz Eliezer.
Rav Moshe begins by describing the general situation of adoption, noting that it is a wonderful thing for a couple to do to adopt an orphan, and that often times this is done because the couple themselves have been unsuccessful in having children and adoption gives them the opportunity to raise a Jewish child. He further notes that it is important to eventually let the children know that they are adopted so that they do not have problems later in life in terms of potentially marrying their biological relatives, and, as such, it is important to try to find out who their real parents are or were (assuming the child is Jewish from birth).
In terms of the yichud issue, Rav Moshe rules that it is not a problem. His primary prooftext is a Gemara in Sotah 43 that rules that an adopted daughter (or step-daughter) should not marry her step-siblings since even though they are not blood relations, it presents the appearance of siblings marrying one another. Based on this, Rav Moshe extrapolates that if the father would be careful to not have yichud with this daughter, it would "blow their cover" and make it clear that she was not actually related. As such, it must be that no prohibition of yichud is introduced in such a case.
Rav Moshe further notes that yichud is not a problem since the parent would not dare do anything untoward with the stepchild or adopted child, insofar as he or she fears his or her spouse finding out. As such, Rav Moshe cautions that if the parent of the same gender as the child passes away, then the remaining parent should try to avoid yichud situations going forward.