Square-free graphs with no induced fork
Electronic Journal of Combinatorics
The claw is the graph K1,3, and the fork is the graph obtained from the claw K1,3 by subdividing one of its edges once. In this paper, we prove a structure theorem for the class of (claw, C4)-free graphs that are not quasi-line graphs, and a structure theorem for the class of (fork, C4)-free graphs that uses the class of (claw, C4)-free graphs as a basic class. Finally, we show that every (fork, C4)-free graph G satisfies (formula presented) via these structure theorems with some additional work on coloring basic classes.
Chudnovsky, Maria; Huang, Shenwei; Karthick, T.; and Kaufmann, Jenny, "Square-free graphs with no induced fork" (2021). Journal Articles. 2197.