25 research outputs found
Partitioning chordal graphs into independent sets and cliques
We consider the following generalization of split graphs: A graph is said to be a (k,ℓ)-graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k=ℓ=1.) Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k,ℓ)-graphs in general. (For instance, being a (k,0)-graph is equivalent to being k-colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal (k,ℓ)-graphs. We also give an O(n(m+n)) recognition algorithm for chordal (k,ℓ)-graphs. When k=1, our algorithm runs in time O(m+n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list’ (or ‘pre-colouring extension’) version of the split partition problem — given a graph with some vertices pre-assigned to the independent set, or to the clique, is there a split partition extending this pre-assignment? Another way to think of our main result is the following min-max property of chordal graphs: the maximum number of independent (i.e., disjoint and nonadjacent) Kr's equals the minimum number of cliques that meet all Kr's
Parameterized Algorithms on Perfect Graphs for deletion to -graphs
For fixed integers , a graph is called an {\em
-graph} if the vertex set can be partitioned into
independent sets and cliques. The class of graphs
generalizes -colourable graphs (when and hence not surprisingly,
determining whether a given graph is an -graph is \NP-hard even when
or in general graphs.
When and are part of the input, then the recognition problem is
NP-hard even if the input graph is a perfect graph (where the {\sc Chromatic
Number} problem is solvable in polynomial time). It is also known to be
fixed-parameter tractable (FPT) on perfect graphs when parameterized by and
. I.e. there is an f(r+\ell) \cdot n^{\Oh(1)} algorithm on perfect
graphs on vertices where is some (exponential) function of and
.
In this paper, we consider the parameterized complexity of the following
problem, which we call {\sc Vertex Partization}. Given a perfect graph and
positive integers decide whether there exists a set of size at most such that the deletion of from results in an
-graph. We obtain the following results: \begin{enumerate} \item {\sc
Vertex Partization} on perfect graphs is FPT when parameterized by .
\item The problem does not admit any polynomial sized kernel when parameterized
by . In other words, in polynomial time, the input graph can not be
compressed to an equivalent instance of size polynomial in . In fact,
our result holds even when .
\item When are universal constants, then {\sc Vertex Partization} on
perfect graphs, parameterized by , has a polynomial sized kernel.
\end{enumerate
Dichotomy for tree-structured trigraph list homomorphism problems
Trigraph list homomorphism problems (also known as list matrix partition
problems) have generated recent interest, partly because there are concrete
problems that are not known to be polynomial time solvable or NP-complete. Thus
while digraph list homomorphism problems enjoy dichotomy (each problem is
NP-complete or polynomial time solvable), such dichotomy is not necessarily
expected for trigraph list homomorphism problems. However, in this paper, we
identify a large class of trigraphs for which list homomorphism problems do
exhibit a dichotomy. They consist of trigraphs with a tree-like structure, and,
in particular, include all trigraphs whose underlying graphs are trees. In
fact, we show that for these tree-like trigraphs, the trigraph list
homomorphism problem is polynomially equivalent to a related digraph list
homomorphism problem. We also describe a few examples illustrating that our
conditions defining tree-like trigraphs are not unnatural, as relaxing them may
lead to harder problems
On Split-Coloring Problems
We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this are