On Convexity in Split graphs: Complexity of Steiner tree and Domination

Given a graph $G$ with a terminal set $R \subseteq V(G)$, the Steiner tree problem (STREE) asks for a set $S\subseteq V(G) \setminus R$ such that the graph induced on $S\cup R$ is connected. A split graph is a graph which can be partitioned into a clique and an independent set. It is known that STREE is NP-complete on split graphs \cite{white1985steiner}. To strengthen this result, we introduce convex ordering on one of the partitions (clique or independent set), and prove that STREE is polynomial-time solvable for tree-convex split graphs with convexity on clique ($K$), whereas STREE is NP-complete on tree-convex split graphs with convexity on independent set ($I$). We further strengthen our NP-complete result by establishing a dichotomy which says that for unary-tree-convex split graphs (path-convex split graphs), STREE is polynomial-time solvable, and NP-complete for binary-tree-convex split graphs (comb-convex split graphs). We also show that STREE is polynomial-time solvable for triad-convex split graphs with convexity on $I$, and circular-convex split graphs. Further, we show that STREE can be used as a framework for the dominating set problem (DS) on split graphs, and hence the classical complexity (P vs NPC) of STREE and DS is the same for all these subclasses of split graphs. Furthermore, it is important to highlight that in \cite{CHLEBIK20081264}, it is incorrectly claimed that the problem of finding a minimum dominating set on split graphs cannot be approximated within $(1-\epsilon)\ln |V(G)|$ in polynomial-time for any $\epsilon >0$ unless NP $\subseteq$ DTIME $n^{O(\log \log n)}$. When the input is restricted to split graphs, we show that the minimum dominating set problem has $2-\frac{1}{|I|}$-approximation algorithm that runs in polynomial time.