11 research outputs found
Linear-time Algorithms for Eliminating Claws in Graphs
Since many NP-complete graph problems have been shown polynomial-time
solvable when restricted to claw-free graphs, we study the problem of
determining the distance of a given graph to a claw-free graph, considering
vertex elimination as measure. CLAW-FREE VERTEX DELETION (CFVD) consists of
determining the minimum number of vertices to be removed from a graph such that
the resulting graph is claw-free. Although CFVD is NP-complete in general and
recognizing claw-free graphs is still a challenge, where the current best
algorithm for a graph has the same running time of the best algorithm for
matrix multiplication, we present linear-time algorithms for CFVD on weighted
block graphs and weighted graphs with bounded treewidth. Furthermore, we show
that this problem can be solved in linear time by a simpler algorithm on
forests, and we determine the exact values for full -ary trees. On the other
hand, we show that CLAW-FREE VERTEX DELETION is NP-complete even when the input
graph is a split graph. We also show that the problem is hard to approximate
within any constant factor better than , assuming the Unique Games
Conjecture.Comment: 20 page
Solving the weighted stable set problem in claw-free graphs via decomposition
We propose an algorithm for solving the maximum weighted stable set problem on claw-free graphs that runs in O(|V|(|E| + |V| log|V|))-time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present article. Despite being weaker than the structural results for claw-free graphs given by Chudnovsky and Seymour [2005, 2008a, 2008b] our decomposition theorem is, on the other hand, algorithmic, that is, it is coupled with an O(|V||E|)-time algorithm that actually produces the decomposition