http://en.wikipedia.org/wiki/Family_life_and_children_of_Vladimir_I :

Anna Porphyrogeneta, daughter of Emperor Romanos II and Theophano, was the only princess of the Makedones to have been married to a foreigner. The Byzantine emperors regarded the Franks and Russians as barbarians, refusing Hugues Capet's proposals to marry Anna to his son Robert I, so the Baptism of Kievan Rus was a prerequisite for this marriage. Following the wedding, Vladimir is said to have divorced all his pagan wives, although this claim is disputed. Regarded by later Russians as a saint, Anna was interred with her husband in the Church of the Tithes....

There is also a case for Yaroslav's descent from Anna. According to this theory, Nestor the Chronicler deliberately represented Yaroslav as Rogneda's son, because he systematically removed all information concerning Kievan ties with Byzantium, spawning pro-Varangian bias (see Normanist theory for details). Proponents allege that Yaroslav's true age was falsified by Nestor, who attempted to represent him as 10 years older than he actually had been, in order to justify Yaroslav's seizure of the throne at the expense of his older brothers.

The Primary Chronicle, for instance, states that Yaroslav died at the age of 76 in 1054 (thus putting his birth at 978), while dating Vladimir's encounter and marriage to Yaroslav's purported mother, Rogneda, to 980. Elsewhere, speaking about Yaroslav's rule in Novgorod (1016), Nestor says that Yaroslav was 28, thus putting his birth at 988. The forensic analysis of Yaroslav's skeleton seems to have confirmed these suspicions, estimating Yaroslav's birth at ca. 988-990, after both the Baptism of Kievan Rus and Vladimir's divorce of Rogneda. Consequently, it is assumed that Yaroslav was either Vladimir's natural son born after the latter's baptism or his son by Anna.

Had Yaroslav an imperial Byzantine descent, he likely would not have stinted to advertise it. Some have seen the willingness of European kings to marry Yaroslav's daughters as an indication of this imperial descent. Subsequent Polish chroniclers and historians, in particular, were eager to view Yaroslav as Anna's son. Recent proponents invoke onomastic arguments, which have often proven decisive in the matters of medieval prosopography. It is curious that Yaroslav named his elder son Vladimir (after his own father) and his eldest daughter Anna (as if after his own mother). Also, there is a certain pattern in his sons having Slavic names (as Vladimir), and his daughters having Greek names only (as Anna). However, in the absence of better sources, Anna's maternity remains a pure speculation.

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Arlogia's mother could be Anna Lekapene of Byzantine - sources needed.

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Children of Milolika Prinsessa av Bulgarien och St Vladimir storfursten i Kiev är:

Boris Vladimirovitj David Duke of Rostov föddes 0982 i Rostov, Yaroslavl, Ryssland, och dog 25 juli, 1015.

Gleb Roman Vladimirovitj hertig av Murom föddes 0983 i Murom, Vladimir, Ryssland, och dog 5 augusti 1015.

Stanslav Vladimirovitj hertig av Smolensk föddes 0984 i Smolensk, Ryssland, och dog 1015.

Pozvizd Vladimirovitj hertig av Vladimir Volynsk föddes 0985 i Vladimir, Volynskij, Volyn, Ukraina, och dog 1015. 

Sudislav Vladimirovitj Duke of Pskov föddes 0986 i Psovk, Ryssland, och dog 1063. 

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http://en.wikipedia.org/wiki/Family_life_and_children_of_Vladimir_I#Anna_Porphyrogeneta

Anna Porphyrogeneta

Anna Porphyrogeneta, daughter of Emperor Romanos II and Theophano, was the only princess of the Makedones to have been married to a foreigner. The Byzantine emperors regarded the Franks and Russians as barbarians, refusing Hugues Capet's proposals to marry Anna to his son Robert I, so the Baptism of Kievan Rus was a prerequisite for this marriage. Following the wedding, Vladimir is said to have divorced all his pagan wives, although this claim is disputed. Regarded by later Russians as a saint, Anna was interred with her husband in the Church of the Tithes.

Anna is not known to have had any children. Either her possible barrenness or the Byzantine house rule could account for this. Had she had any progeny, the prestigious and much sought imperial parentage would have certainly been advertised by her descendants. Hagiographic sources, contrary to the Primary Chronicle, posit Boris and Gleb as her offspring, on the understanding that holy brothers should have had a holy mother.

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http://en.wikipedia.org/wiki/Vladimir_I_of_Kiev#Family

• Anna Porphyrogeneta

o Theofana, a wife of Novgorod posadnik Ostromir, a grandson of semi-legendary Dobrynya (highly doubtful is the fact of her being Anna's offspring)

-------------------- http://en.wikipedia.org/wiki/Family_life_and_children_of_Vladimir_I#Greek_wife

Greek wife

During his unruly youth, Vladimir begot his eldest son, Sviatopolk, relations with whom would cloud his declining years. His mother was a Greek nun captured by Svyatoslav I in Bulgaria and married to his lawful heir Yaropolk I. Russian historian Vasily Tatischev, invariably erring in the matters of onomastics, gives her the fanciful Roman name of Julia. When Yaropolk was murdered by Vladimir's agents, the new sovereign raped his wife and she soon (some[who?] would say, too soon) gave birth to a child. Thus, Sviatopolk was probably the eldest of Vladimir's sons, although the issue of his parentage has been questioned and he has been known in the family as "the son of two fathers"