Off Broadway’s Public Theater has added the new tuner adaptation of “Fun Home” to the fall lineup of its Public Lab series of smaller-scale productions of developing titles.Judy Kuhn, Roberta Colindrez, Beth Malone and Joel Perez make up the cast of the show, an adaptation of the graphic memoir by Alison Bechdel. Sam Gold (“Seminar”) directs. “Fun Home,” which has music by Jeanine Tesori and book and lyrics by Lisa Kron, recently was developed as part of the Sundance Theater Institute’s 2012 Theater Lab. Both Tesori and Kron have prior relationships with the Public, with the company having bowed the composer’s tuner “Caroline, or Change” (written with Tony Kushner) as well as past Kron plays including “Well” and “In the Wake.” Set in a family-owned funeral home, the storyline traces a woman’s complicated relationship with her late father. Malone plays Alison, the character based on Bechdel. “Fun Home” joins a Public Lab slate that also includes Richard Nelson’s “Sorry” and Dominique Morisseau’s “Detroit ’67.” “Fun Home” runs Oct. 17-Nov. 4 in the newly renovated Public Theater outpost in downtown Gotham.