Constrained Hitting Set and Steiner Tree in $SC_k$ and $2K_2$-free Graphs

\emph{Strictly Chordality-$k$ graphs ($SC_k$)} are graphs which are either cycle-free or every induced cycle is of length exactly $k, k \geq 3$. Strictly chordality-3 and strictly chordality-4 graphs are well known chordal and chordal bipartite graphs, respectively. For $k\geq 5$, the study has been recently initiated in \cite{sadagopan} and various structural and algorithmic results are reported. In this paper, we show that maximum independent set (MIS), minimum vertex cover, minimum dominating set, feedback vertex set (FVS), odd cycle transversal (OCT), even cycle transversal (ECT) and Steiner tree problem are polynomial time solvable on $SC_k$ graphs, $k\geq 5$. We next consider $2K_2$-free graphs and show that FVS, OCT, ECT, Steiner tree problem are polynomial time solvable on subclasses of $2K_2$-free graphs.