From The Legend of the Lamed Vovniks
The Jewish legend of the Lamed Vovniks describes thirty-six righteous men in every generation upon whose merit the world is kept from entire destruction. Based in part on the story of Abraham and his conversation with the Lord about the destruction of Sodom in Genesis 18, the Lamed Vovniks are those who, by virtue of their compassion for others and the prayers they offer, cause the Lord to answer, “I will spare all the place for their sakes” (Genesis 18:26).
Yes, that seems like the same story, though I had never heard the terms Lamed-vav Tsadikim or Yiddish Lamedvavniks.
Must be the story is told several differnt ways.
I do like to think that Linda Gibbons, and a few others I have heard of, are among these Secret Tzaddiks.