13 research outputs found
On minimum -claw deletion in split graphs
For , is called -claw. In minimum -claw deletion
problem (\texttt{Min--Claw-Del}), given a graph , it is required
to find a vertex set of minimum size such that is
-claw free. In a split graph, the vertex set is partitioned into two sets
such that one forms a clique and the other forms an independent set. Every
-claw in a split graph has a center vertex in the clique partition. This
observation motivates us to consider the minimum one-sided bipartite -claw
deletion problem (\texttt{Min--OSBCD}). Given a bipartite graph , in \texttt{Min--OSBCD} it is asked to find a vertex set of
minimum size such that has no -claw with the center
vertex in . A primal-dual algorithm approximates \texttt{Min--OSBCD}
within a factor of . We prove that it is \UGC-hard to approximate with a
factor better than . We also prove it is approximable within a factor of 2
for dense bipartite graphs. By using these results on \texttt{Min--OSBCD},
we prove that \texttt{Min--Claw-Del} is \UGC-hard to approximate within a
factor better than , for split graphs. We also consider their complementary
maximization problems and prove that they are \APX-complete.Comment: 11 pages and 1 figur
On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion
In this paper, we investigate the approximability of two node deletion
problems. Given a vertex weighted graph and a specified, or
"distinguished" vertex , MDD(min) is the problem of finding a minimum
weight vertex set such that becomes the
minimum degree vertex in ; and MDD(max) is the problem of
finding a minimum weight vertex set such that
becomes the maximum degree vertex in . These are known
-complete problems and have been studied from the parameterized complexity
point of view in previous work. Here, we prove that for any ,
both the problems cannot be approximated within a factor , unless . We also show that for any
, MDD(min) cannot be approximated within a factor on bipartite graphs, unless , and that for any , MDD(max) cannot be approximated within a
factor on bipartite graphs, unless . We give an factor approximation algorithm
for MDD(max) on general graphs, provided the degree of is . We
then show that if the degree of is , a similar result holds
for MDD(min). We prove that MDD(max) is -complete on 3-regular unweighted
graphs and provide an approximation algorithm with ratio when is a
3-regular unweighted graph. In addition, we show that MDD(min) can be solved in
polynomial time when is a regular graph of constant degree.Comment: 16 pages, 4 figures, submitted to Elsevier's Journal of Discrete
Algorithm
On the Complexity of Co-secure Dominating Set Problem
A set of a graph is a dominating set of if
every vertex is adjacent to at least one vertex in A
set is a co-secure dominating set (CSDS) of a graph if
is a dominating set of and for each vertex there exists a vertex
such that and
is a dominating set of . The minimum cardinality of a co-secure dominating
set of is the co-secure domination number and it is denoted by
. Given a graph , the minimum co-secure dominating
set problem (Min Co-secure Dom) is to find a co-secure dominating set of
minimum cardinality. In this paper, we strengthen the inapproximability result
of Min Co-secure Dom for general graphs by showing that this problem can not be
approximated within a factor of for perfect elimination
bipartite graphs and star convex bipartite graphs unless P=NP. On the positive
side, we show that Min Co-secure Dom can be approximated within a factor of
for any graph with . For -regular and
-regular graphs, we show that Min Co-secure Dom is approximable within a
factor of and , respectively. Furthermore, we
prove that Min Co-secure Dom is APX-complete for -regular graphs.Comment: 12 pages, 2 figure
The Complexity of Finding Subgraphs Whose Matching Number Equals the Vertex Cover Number
The class of graphs where the size of a minimum vertex cover equals that of a maximum matching is known as König-Egerváry graphs. König-Egerváry graphs have been studied extensively from a graph theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding maximum König-Egerváry subgraphs of a given graph. More specifically, we look at the problem of finding a minimum number of vertices or edges to delete to make the resulting graph König-Egerváry. We show that both these versions are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. En route, we point out an interesting connection between the vertex deletion version and the Above Guarantee Vertex Cover problem where one is interested in the parameterized complexity of the Vertex Cover problem when parameterized by the ‘additional number of vertices ’ needed beyond the matching size. This connection is of independent interest and could be useful in establishing the parameterized complexity of Above Guarantee Vertex Cover problem
The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover
A graph is König-Egerváry if the size of a minimum vertex cover equals that of a maximum matching in the graph. These graphs have been studied extensively from a graph-theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding König-Egerváry subgraphs of a given graph. In particular, given a graph G and a nonnegative integer k, we are interested in the following questions: 1. does there exist a set of k vertices (edges) whose deletion makes the graph König-Egerváry? 2. does there exist a set of k vertices (edges) that induce a König-Egerváry subgraph? We show that these problems are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. Towards this end, we first study the algorithmic complexity of Above Guarantee Vertex Cover, where one is interested in minimizing the additional number of vertices needed beyond the maximum matching size for the vertex cover. Further, while studying the parameterized complexity of the problem of deleting k vertices to obtain a König-Egerváry graph, we show a number of interesting structural results on matchings and vertex covers which could be useful in other contexts