3,453 research outputs found
Even and odd pairs in comparability and in P4-comparability graphs
AbstractWe characterize even and odd pairs in comparability and in P4-comparability graphs. The characterizations lead to simple algorithms for deciding whether a given pair of vertices forms an even or odd pair in these classes of graphs. The complexities of the proposed algorithms are O(n + m) for comparability graphs and O(n2m) for P4-comparability graphs. The former represents an improvement over a recent algorithm of complexity O(nm)
Infinite cographs and chain complete N-free posets
We give a necessary and sufficient condition for a -free graph to be a
cograph. This allows us to obtain a simple proof of the fact that finite
-free graphs are finite cographs. We also prove that chain complete posets
whose comparability graph is a cograph are series-parallel.Comment: 7 page
Strong Koszulness of toric rings associated with stable set polytopes of trivially perfect graphs
We give necessary and sufficient conditions for strong Koszulness of toric
rings associated with the stable set polytope of graphs.Comment: 10 pages, 7 figures, 1 table, to appear in Journal of Algebra and its
Application
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
Linear Time LexDFS on Cocomparability Graphs
Lexicographic depth first search (LexDFS) is a graph search protocol which
has already proved to be a powerful tool on cocomparability graphs.
Cocomparability graphs have been well studied by investigating their
complements (comparability graphs) and their corresponding posets. Recently
however LexDFS has led to a number of elegant polynomial and near linear time
algorithms on cocomparability graphs when used as a preprocessing step [2, 3,
11]. The nonlinear runtime of some of these results is a consequence of
complexity of this preprocessing step. We present the first linear time
algorithm to compute a LexDFS cocomparability ordering, therefore answering a
problem raised in [2] and helping achieve the first linear time algorithms for
the minimum path cover problem, and thus the Hamilton path problem, the maximum
independent set problem and the minimum clique cover for this graph family
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
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