1 research outputs found
Linear Recognition of Almost Interval Graphs
Let \mbox{interval} + k v, \mbox{interval} + k e, and \mbox{interval} -
k e denote the classes of graphs that can be obtained from some interval graph
by adding vertices, adding edges, and deleting edges, respectively.
When is small, these graph classes are called almost interval graphs. They
are well motivated from computational biology, where the data ought to be
represented by an interval graph while we can only expect an almost interval
graph for the best. For any fixed , we give linear-time algorithms for
recognizing all these classes, and in the case of membership, our algorithms
provide also a specific interval graph as evidence. When is part of the
input, these problems are also known as graph modification problems, all
NP-complete. Our results imply that they are fixed-parameter tractable
parameterized by , thereby resolving the long-standing open problem on the
parameterized complexity of recognizing \mbox{interval}+ k e, first asked by
Bodlaender et al. [Bioinformatics, 11:49--57, 1995]. Moreover, our algorithms
for recognizing \mbox{interval}+ k v and \mbox{interval}- k e run in times
and , (where and stand for
the numbers of vertices and edges respectively in the input graph,)
significantly improving the -time algorithm of Heggernes
et al. [STOC 2007] and the -time algorithm of Cao and Marx
[SODA 2014] respectively.Comment: Completely restructured, and results on unit interval graphs have
been dropped to make this version more focuse