62 research outputs found
Maximal cliques structure for cocomparability graphs and applications
Il s'agit d'une recherche sur les relations entre les graphes d'intervalles et les graphes de cocomparabilitéA cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. In this paper we investigate the relationships between interval and cocomparabil-ity graphs. This study is motivated by recent results [5, 13] that show that for some problems, the algorithm used on interval graphs can also be used with small modifications on cocomparability graphs. Many of these algorithms are based on graph searches that preserve cocomparability orderings. First we propose a characterization of cocomparability graphs via a lattice structure on the set of their maximal cliques. Using this characterization we can prove that every maximal interval subgraph of a cocomparability graph G is also a maximal chordal subgraph of G. Although the size of this lattice of maximal cliques can be exponential in the size of the graph, it can be used as a framework to design and prove algorithms on cocomparability graphs. In particular we show that a new graph search, namely Local Maximal Neighborhood Search (LocalMNS) leads to an O(n + mlogn) time algorithm to find a maximal interval subgraph of a cocomparability graph. Similarly we propose a linear time algorithm to compute all simplicial vertices in a cocomparability graph. In both cases we improve on the current state of knowledge
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
Graphs with at most two moplexes
A moplex is a natural graph structure that arises when lifting Dirac's
classical theorem from chordal graphs to general graphs. However, while every
non-complete graph has at least two moplexes, little is known about structural
properties of graphs with a bounded number of moplexes. The study of these
graphs is motivated by the parallel between moplexes in general graphs and
simplicial modules in chordal graphs: Unlike in the moplex setting, properties
of chordal graphs with a bounded number of simplicial modules are well
understood. For instance, chordal graphs having at most two simplicial modules
are interval. In this work we initiate an investigation of -moplex graphs,
which are defined as graphs containing at most moplexes. Of particular
interest is the smallest nontrivial case , which forms a counterpart to
the class of interval graphs. As our main structural result, we show that the
class of connected -moplex graphs is sandwiched between the classes of
proper interval graphs and cocomparability graphs; moreover, both inclusions
are tight for hereditary classes. From a complexity theoretic viewpoint, this
leads to the natural question of whether the presence of at most two moplexes
guarantees a sufficient amount of structure to efficiently solve problems that
are known to be intractable on cocomparability graphs, but not on proper
interval graphs. We develop new reductions that answer this question negatively
for two prominent problems fitting this profile, namely Graph Isomorphism and
Max-Cut. On the other hand, we prove that every connected -moplex graph
contains a Hamiltonian path, generalising the same property of connected proper
interval graphs. Furthermore, for graphs with a higher number of moplexes, we
lift the previously known result that graphs without asteroidal triples have at
most two moplexes to the more general setting of larger asteroidal sets
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
Feedback vertex set on chordal bipartite graphs
Let G=(A,B,E) be a bipartite graph with color classes A and B. The graph G is
chordal bipartite if G has no induced cycle of length more than four. Let
G=(V,E) be a graph. A feedback vertex set F is a set of vertices F subset V
such that G-F is a forest. The feedback vertex set problem asks for a feedback
vertex set of minimal cardinality. We show that the feedback vertex set problem
can be solved in polynomial time on chordal bipartite graphs
On treewidth and minimum fill-in of asteroidal triple-free graphs
We present O(n5R + n3R3) time algorithms to compute the treewidth, pathwidth, minimum fill-in and minimum interval graph completion of asteroidal triple-free graphs, where n is the number of vertices and R is the number of minimal separators of the input graph. This yields polynomial time algorithms for the four NP-complete graph problems on any subclass of the asteroidal triple-free graphs that has a polynomially bounded number of minimal separators, as e.g. cocomparability graphs of bounded dimension and d-trapezoid graphs for any fixed d ⩾ 1
Optimal Linear Arrangement of Interval Graphs
http://www.springerlink.com/content/wrk8346657167528/?p=a1f283d79ac14affb4d165501b84e569&pi=23We study the optimal linear arrangement (OLA) problem on interval graphs. Several linear layout problems that are NP-hard on general graphs are solvable in polynomial time on interval graphs.We prove that, quite surprisingly, optimal linear arrangement of interval graphs is NP-hard. The same result holds for permutation graphs. We present a lower bound and a simple and fast 2-approximation algorithm based on any interval model of the input graph
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