Given a connected graph $G$ and a terminal set $R subseteq V(G)$, Steiner tree asks for a tree that includes all of R with at most $r$ edges for some integer $r ge 0$. It is known from [ND12,Garey et. al [1]] that Steiner tree is NP-complete in general graphs. Split graph is a graph which can be partitioned into a clique and an independent set. K. White et. al [2] has established that Steiner tree in split graphs is NP-complete. In this talk, we present an interesting dichotomy: we show that Steiner tree on $K_{1,4}$-free split graphs is polynomial-time solvable, whereas, Steiner tree on $K_{1,5}$-free split graphs is NP-complete. We investigate $K_{1,4}$-free and $K_{1,3}$-free (also known as claw-free) split graphs from a structural perspective. Further, using our structural study, we present polynomial-time algorithms for Steiner tree in $K_{1,4}$-free and $K_{1,3}$-free split graphs.
