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

TL;DR

This paper demonstrates that several classical graph problems, including Steiner tree and feedback vertex set, are solvable in polynomial time on certain classes of graphs, specifically $SC_k$ and $2K_2$-free graphs, expanding the understanding of their computational complexity.

Contribution

It establishes polynomial-time algorithms for multiple problems on $SC_k$ graphs for $k \\geq 5$ and on subclasses of $2K_2$-free graphs, extending known results.

Findings

Polynomial-time solvability of MIS, MVC, MDS, FVS, OCT, ECT, Steiner tree on $SC_k$ graphs for $k \\geq 5$.

Polynomial-time algorithms for FVS, OCT, ECT, Steiner tree on subclasses of $2K_2$-free graphs.

Advances the understanding of algorithmic complexity in special graph classes.

Abstract

\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.