Short Cycles Dictate Dichotomy Status of the Steiner Tree Problem on Bisplit Graphs

A graph G is said to be a bisplit graph if its vertex set can be partitioned into a stable set and a complete bipartite graph. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph G and a subset of vertices R. V (G), the objective is to find a minimum cardinality set S. V (G) such that the set R. S induces a connected subgraph. In this paper, we present an interesting dichotomy result for STREE on bisplit graphs, we show that STREE is polynomial-time solvable for chordal bipartite bisplit graphs, and NP-complete otherwise. Further, we show that for chordal bisplit graphs, the problem is polynomial-time solvable. A revisit of our NP-complete reduction instances reveals that the instances are diameter at most 5 bipartite graphs. We also obtain one more dichotomy result for STREE on bisplit graphs which says that for diameter 5 the problem is NP-complete and polynomial-time solvable for diameter at most 4. On the parameterized complexity front, we show that the parameterized version of Steiner tree problem on bisplit graphs is fixed-parameter tractable when the parameter is the biclique size and is W[2]-hard on bisplit graphs if the parameter is the solution size. We conclude this paper by presenting structural results of bisplit graphs which will be of use to solve other combinatorial problems.