1,001,941 research outputs found
On graph equivalences preserved under extensions
Let R be an equivalence relation on graphs. By the strengthening of R we mean
the relation R' such that graphs G and H are in the relation R' if for every
graph F, the union of the graphs G and F is in the relation R with the union of
the graphs H and F. We study strengthenings of equivalence relations on graphs.
The most important case that we consider concerns equivalence relations defined
by graph properties. We obtain results on the strengthening of equivalence
relations determined by the properties such as being a k-connected graph,
k-colorable, hamiltonian and planar
On graphs with representation number 3
A graph is word-representable if there exists a word over the
alphabet such that letters and alternate in if and only if
is an edge in . A graph is word-representable if and only if it is
-word-representable for some , that is, if there exists a word containing
copies of each letter that represents the graph. Also, being
-word-representable implies being -word-representable. The minimum
such that a word-representable graph is -word-representable, is called
graph's representation number.
Graphs with representation number 1 are complete graphs, while graphs with
representation number 2 are circle graphs. The only fact known before this
paper on the class of graphs with representation number 3, denoted by
, is that the Petersen graph and triangular prism belong to this
class. In this paper, we show that any prism belongs to , and
that two particular operations of extending graphs preserve the property of
being in . Further, we show that is not included
in a class of -colorable graphs for a constant . To this end, we extend
three known results related to operations on graphs.
We also show that ladder graphs used in the study of prisms are
-word-representable, and thus each ladder graph is a circle graph. Finally,
we discuss -word-representing comparability graphs via consideration of
crown graphs, where we state some problems for further research
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
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
- …