Dichotomy study of the Steiner tree problem in split-like graphs

Given a connected graph $G$ and a terminal set $R \subseteq V(G)$, the minimum Steiner tree problem (ST) asks for a tree that spans all of $R$ with at most $r$ vertices from $V(G)\backslash R$, for some integer $r\geq 0$. A \emph{split graph} is a graph which can be partitioned into a clique and an independent set. It is known from (Garey et al.,1977) that ST is NP-complete, even for split graphs . We introduce the class of split-like graphs which unifies several known graph classes like bipartite graphs, split graphs, and bisplit graphs, allowing for a cohesive study across multiple structural constraints. We investigate the computational complexity of the Steiner tree problem under structural constraints, specifically $K_{1,r}$-free, bounded diameter, chordality and star-convexity. Through reductions (primarily from Exact-3-Cover and its variants), the paper establishes a series of dichotomy results. It precisely gives the boundary for $K_{1,r}$-free bipartite graphs: ST is in P for $r \le 3$ and NP-complete for $r \ge 4$; whereas on $K_{1,r}$-free bisplit graphs, ST is in P for any fixed $r\geq 3$. On bisplit graphs, the Steiner tree problem admits a polynomial-time solution when the diameter is 2. In contrast, for diameters 3 and 4, the problem is NP-complete. The problem is NP-complete under star convexity constraints on the independent set. When star convexity is imposed on the $k$-clique side, the problem is solvable in polynomial time. The problem is NP-complete on chordal bipartite graphs and chordal split graphs (i.e., split graphs themselves), while polynomial-time algorithms exist for other subclasses of split-like graphs.