7 research outputs found
Linear Time LexDFS on Chordal Graphs
Lexicographic Depth First Search (LexDFS) is a special variant of a Depth
First Search (DFS), which was introduced by Corneil and Krueger in 2008. While
this search has been used in various applications, in contrast to other graph
searches, no general linear time implementation is known to date. In 2014,
K\"ohler and Mouatadid achieved linear running time to compute some special
LexDFS orders for cocomparability graphs. In this paper, we present a linear
time implementation of LexDFS for chordal graphs. Our algorithm is able to find
any LexDFS order for this graph class. To the best of our knowledge this is the
first unrestricted linear time implementation of LexDFS on a non-trivial graph
class. In the algorithm we use a search tree computed by Lexicographic Breadth
First Search (LexBFS)
Finding Biclique Partitions of Co-Chordal Graphs
The biclique partition number of a graph is referred to as
the least number of complete bipartite (biclique) subgraphs that are required
to cover the edges of the graph exactly once. In this paper, we show that the
biclique partition number () of a co-chordal (complementary graph of
chordal) graph is less than the number of maximal cliques
() of its complementary graph: a chordal graph . We
first provide a general framework of the ``divide and conquer" heuristic of
finding minimum biclique partitions of co-chordal graphs based on clique trees.
Furthermore, a heuristic of complexity is proposed by
applying lexicographic breadth-first search to find structures called moplexes.
Either heuristic gives us a biclique partition of with size
. In addition, we prove that both of our heuristics can solve
the minimum biclique partition problem on exactly if its complement
is chordal and clique vertex irreducible. We also show that if is a split graph
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
A simple algorithm to generate the minimal separators and the maximal cliques of a chordal graph
We present a simple unified algorithmic process which uses either LexBFS or MCS on a chordal graph to generate the minimal separators and the maximal cliques in linear time in a single pass
Organizing the atoms of the clique separator decomposition into an atom tree
International audienceWe define an atom tree of a graph as a generalization of a clique tree: its nodes are the atoms obtained by clique minimal separator decomposition, and its edges correspond to the clique minimal separators of the graph.Given a graph GG, we compute an atom tree by using a clique tree of a minimal triangulation HH of GG. Computing an atom tree with such a clique tree as input can be done in O(min(nm,m+nf))O(min(nm,m+nf)), where ff is the number of fill edges added by the triangulation. When both a minimal triangulation and the clique minimal separators of GG are provided, we compute an atom tree of GG in O(m+f)O(m+f) time, which is in O(n2)O(n2) time.We give an O(nm)O(nm) time algorithm, based on MCS, which combines in a single pass the 3 steps involved in building an atom tree: computing a minimal triangulation, constructing a clique tree, and constructing the corresponding atom tree.Finally, we present a process which uses a traversal of a clique tree of a minimal triangulation to determine the clique minimal separators and build the corresponding atom tree in O(n(n+t))O(n(n+t)) time, where tt is the number of 2-pairs of HH (tt is at most View the MathML sourcem¯−f, where View the MathML sourcem¯ is the number of edges of the complement graph); to complete this, we also give an algorithm which computes a minimal triangulation in View the MathML sourceO(n(n+m¯)) time, thus providing an approach to compute the decomposition in View the MathML sourceO(n(n+m¯)) time
Influence of the tie-break rule on the end-vertex problem
International audienceEnd-vertices of a given graph search may have some nice properties, as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial, see Rose, Tarjan and Lueker 1976. Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not, as first studied in Corneil, Köhler and Lanlignel 2010. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break in case of equality during the search, for a given graph search including the classic ones such as BFS and DFS, can determine the complexity of the end-vertex problem. In particular we prove a counterintuitive NP-completeness result for Breadth First Search solving a problem raised in Corneil, Köhler and Lanlignel 2010