Complexity of Steiner Tree in Split Graphs - Dichotomy Results

Given a connected graph G and a terminal set R ⊆ V (G), Steiner tree asks for a tree that includes all of R with at most r edges for some integer r ≥ 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 paper, we present an interesting dichotomy: we show that Steiner tree on K1,4-free split graphs is polynomial-time solvable, whereas, Steiner tree on K1,5-free split graphs is NP-complete. We investigate K1,4-free and K1,3-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 K1,4-free and K1,3-free split graphs. Although, polynomial-time solvability of K1,3-free split graphs is implied from K1,4-free split graphs, we wish to highlight our structural observations on K1,3-free split graphs which may be used in other combinatorial problems.