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 polynomialtime 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.
CITATION STYLE
Illuri, M., Renjith, P., & Sadagopan, N. (2016). Complexity of Steiner tree in split graphs - Dichotomy results. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9602, pp. 308–325). Springer Verlag. https://doi.org/10.1007/978-3-319-29221-2_27
Mendeley helps you to discover research relevant for your work.