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\emph{Strictly Chordality-$k$ graphs ($SC_k$)} are graphs which are eithercycle-free or every induced cycle is of length exactly $k, k \geq 3$. Strictlychordality-3 and strictly chordality-4 graphs are well known chordal andchordal bipartite graphs, respectively. For $k\geq 5$, the study has beenrecently initiated in \cite{sadagopan} and various structural and algorithmicresults 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 treeproblem are polynomial time solvable on $SC_k$ graphs, $k\geq 5$. We nextconsider $2K_2$-free graphs and show that FVS, OCT, ECT, Steiner tree problemare polynomial time solvable on subclasses of $2K_2$-free graphs.

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