4,281 research outputs found
A New Characterization of Trivially Perfect Graphs
A graph is \emph{trivially perfect} if for every induced subgraph the cardinality of the largest set of pairwise nonadjacent vertices (the stability number) equals the number of (maximal) cliques . We characterize the trivially perfect graphs in terms of vertex-coloring and we extend some definitions to infinite graphs
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page
On the threshold-width of graphs
The GG-width of a class of graphs GG is defined as follows. A graph G has
GG-width k if there are k independent sets N1,...,Nk in G such that G can be
embedded into a graph H in GG such that for every edge e in H which is not an
edge in G, there exists an i such that both endpoints of e are in Ni. For the
class TH of threshold graphs we show that TH-width is NP-complete and we
present fixed-parameter algorithms. We also show that for each k, graphs of
TH-width at most k are characterized by a finite collection of forbidden
induced subgraphs
Chordal Graphs are Fully Orientable
Suppose that D is an acyclic orientation of a graph G. An arc of D is called
dependent if its reversal creates a directed cycle. Let m and M denote the
minimum and the maximum of the number of dependent arcs over all acyclic
orientations of G. We call G fully orientable if G has an acyclic orientation
with exactly d dependent arcs for every d satisfying m <= d <= M. A graph G is
called chordal if every cycle in G of length at least four has a chord. We show
that all chordal graphs are fully orientable.Comment: 11 pages, 1 figure, accepted by Ars Combinatoria (March 26, 2010
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