47 research outputs found
A polynomial kernel for vertex deletion into bipartite permutation graphs
A permutation graph can be defined as an intersection graph of segments whose
endpoints lie on two parallel lines and , one on each. A
bipartite permutation graph is a permutation graph which is bipartite.
In the the bipartite permutation vertex deletion problem we ask for a given
-vertex graph, whether we can remove at most vertices to obtain a
bipartite permutation graph. This problem is NP-complete but it does admit an
FPT algorithm parameterized by .
In this paper we study the kernelization of this problem and show that it
admits a polynomial kernel with vertices
A polynomial kernel for Block Graph Deletion
In the Block Graph Deletion problem, we are given a graph on vertices
and a positive integer , and the objective is to check whether it is
possible to delete at most vertices from to make it a block graph,
i.e., a graph in which each block is a clique. In this paper, we obtain a
kernel with vertices for the Block Graph Deletion problem.
This is a first step to investigate polynomial kernels for deletion problems
into non-trivial classes of graphs of bounded rank-width, but unbounded
tree-width. Our result also implies that Chordal Vertex Deletion admits a
polynomial-size kernel on diamond-free graphs. For the kernelization and its
analysis, we introduce the notion of `complete degree' of a vertex. We believe
that the underlying idea can be potentially applied to other problems. We also
prove that the Block Graph Deletion problem can be solved in time .Comment: 22 pages, 2 figures, An extended abstract appeared in IPEC201
Open problems on graph coloring for special graph classes.
For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring
Towards Constant-Factor Approximation for Chordal / Distance-Hereditary Vertex Deletion
For a family of graphs ?, Weighted ?-Deletion is the problem for which the input is a vertex weighted graph G = (V, E) and the goal is to delete S ? V with minimum weight such that G?S ? ?. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs.
In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when ? is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest