1. Your question is answered by the Gemara in Sanhedrin 72b. The Gemara there cites a Beraisa that states one may kill a burglar who is digging his way into the house in a way that presumably he is prepared to kill, whether he comes on a weekday or on Shabbos. The Gemara states that the reason why it is necessary for the Beraisa to teach that one may kill him even on Shabbos is because otherwise we might have thought that in the same way that Beis Din never puts a person to death on Shabbos, so too one may not kill the Rodef on Shabbos. Rashi (72b, DH Ka) writes that the reason why, according to the conclusion of the Gemara, one may kill the Rodef on Shabbos is the Piku'ach Nefesh of the potential victim. In other words, one is permitted to desecrate Shabbos to save the life of the pursued in the same way that one may drive a dangerously-ill person to the hospital on Shabbos to save his life.

2. This is the simple explanation of the Gemara. I did find a slightly different interpretation in the Me'iri, who writes that even though one does not generally put the convicted to death on Shabbos, in this case one may do so because the Rodef is already considered a Gavra Katila, "a dead man" already.

(One might ask, however, that in every case of a criminal convicted of the death penalty, the criminal is considered a Gavra Ketila, and yet Beis Din does not put him to death on Shabbos. Apparently, the Me'iri's intention is that the Rodef is even more of a total Gavra Ketila, because he must be killed even before he kills his intended victim, and therefore he may be killed even on Shabbos, in contrast to other convicted criminals whose punishment may be postponed to Sunday.)

3. It is possible that the difference in the respective explanations of Rashi and Me'iri may be understood further based on the words of the Noda b'Yehudah (second edition), Choshen Mishpat #60, DH v'Od Ka'amina. He writes about the dispute between Rav Huna and Rav Chisda on Daf 72b. Rav Huna maintains that the law of the Torah that one may kill the Rodef is not a *punishment*, but rather a means of saving the pursued. That is why one may kill even a minor who is pursuing someone. In contrast, Rav Chisda maintains that one kills the Rodef as a *punishment*. Accordingly, one may not kill a Rodef who is a minor, since minors are not subject to punishment. Rashi's explanation is consistent with the understanding of Rav Huna that one kills the Rodef merely to save the life of his victim, and, consequently, there is no problem with doing this on Shabbos. The Me'iri understands that this is also a punishment, and therefore he adds another reason why it may be done on Shabbos -- that the Rodef is considered a Gavra Ketila and therefore one may carry out this punishment even on Shabbos (since one is doing so to save the life of the victim).

D. Bloom