1 research outputs found
Subset Feedback Vertex Set in Chordal and Split Graphs
In the \textsc{Subset Feedback Vertex Set (Subset-FVS)} problem the input is
a graph , a subset of vertices of called the `terminal'
vertices, and an integer . The task is to determine whether there exists a
subset of vertices of cardinality at most which together intersect all
cycles which pass through the terminals. \textsc{Subset-FVS} generalizes
several well studied problems including \textsc{Feedback Vertex Set} and
\textsc{Multiway Cut}. This problem is known to be \NP-Complete even in split
graphs. Cygan et al. proved that \textsc{Subset-FVS} is fixed parameter
tractable (\FPT) in general graphs when parameterized by [SIAM J. Discrete
Math (2013)]. In split graphs a simple observation reduces the problem to an
equivalent instance of the -\textsc{Hitting Set} problem with same solution
size. This directly implies, for \textsc{Subset-FVS} \emph{restricted to split
graphs}, (i) an \FPT algorithm which solves the problem in \OhStar(2.076^k)
time \footnote{The \OhStar() notation hides polynomial factors.}% for
\textsc{Subset-FVS} in Chordal % Graphs [Wahlstr\"om, Ph.D. Thesis], and (ii) a
kernel of size . We improve both these results for
\textsc{Subset-FVS} on split graphs; we derive (i) a kernel of size
which is the best possible unless \NP \subseteq \coNP/{\sf
poly}, and (ii) an algorithm which solves the problem in time
. Our algorithm, in fact, solves \textsc{Subset-FVS} on the
more general class of \emph{chordal graphs}, also in time