One of the most vital sections of the Selihot service is the "Yag Midot," the recitation of the thirteen divine attributes of mercy listed in the Torah (Shemot 34:6-7). The holy books describe the special power of this recitation especially during the month of Elul, the month of repentance when we pray each morning for forgiveness.
There is a debate among the Halachic authorities as to whether the "Yag Middot" may be recited privately, without the presence of a Minyan. The Shulhan Aruch (Orah Haim 565:5) ruled that one may recite the verses of the "Yag Middot" without a Minyan if he reads them with the Te’amim (cantillation notes), as though he reads from the Torah. Even though the "Yag Middot" may not be recited as a prayer without the presence of a Minyan, one may read them privately as though he simply reads verses from the Torah. This is the view followed by the Hid"a (Rav Haim Yosef David Azulai, 1724-1806). The Arizal (Rav Yishak Luria of Safed, 1534-1572), however, held that the section of the "Yag Middot" is so sacred that it may not be read without a Minyan under any circumstances, even if one reads it with the "Te’amim." This was the view adopted by the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), who, when praying Selihot privately, would recite an Aramaic translation of the "Yag Middot" which he found in a work called Keter Malchut. He would not, however, recite the verses in the original Hebrew, even with the "Te’amim."
As for the final Halacha, Hacham Ovadia Yosef writes that common practice follows the lenient ruling of the Shulhan Aruch, and thus one who prays Selihot without a Minyan may read the verses of the "Yag Middot" with the "Te’amim." It should be noted that this applies not only to Selihot, but even on ordinary weekdays when the "Yag Middot" section is recited as part of the Tahanunim. An individual praying privately may recite the "Yag Middot," provided that he reads the text with the "Te’amim."
The Poskim also address the question of whether, according to this view, one must complete the second verse of the "Yag Middot." The list of the thirteen attributes ends with the word "Ve’nakeh," in the middle of a verse; the verse continues with, "Lo Yenakeh Poked Avot Abot Al Banim…" One might argue that if an individual recites these verses as though he reads them from the Torah, and not as prayer, he must complete the entire Pasuk. However, the Poskim conclude that since one is, after all, reciting this verse from the prayer text, he is not required to complete the verse.
Yet another issue arises concerning the paragraph "Kel Melech Yosheb" with which we introduce the Selihot service. This paragraph is written in plural form ("Kel Horeta Lanu…Zechor Lanu Hayom…"), and thus some authorities rule that it may not be recited without the presence of a Minyan. This was indeed the practice of the Ben Ish Hai. The Hid"a, however, disagreed, and ruled that the recitation of this introductory paragraph does not require a Minyan. Hacham Ovadia Yosef, writing in Kol Sinai, followed the Hid"a’s opinion, and ruled that one may recite "Kel Melech Yosheb" without a Minyan, despite the plural form, since this is the standard prayer text.
Summary: It is permissible to recite the "Yag Midot" section (the thirteen attributes of divine mercy) without a Minyan, provided that one reads the verses with their "Te’amim." One may also recite the paragraph of "Kel Melech Yosheb," which introduces the "Yag Middot," without a Minyan.