65,725 research outputs found
A note on minimal matching covered graphs
A graph is called matching covered if for its every edge there is a maximum
matching containing it. It is shown that minimal matching covered graphs
contain a perfect matching.Comment: 4 page
Pure simplicial complexes and well-covered graphs
A graph is called well-covered if all maximal independent sets of
vertices have the same cardinality. A simplicial complex is called
pure if all of its facets have the same cardinality. Let be the
class of graphs with some disjoint maximal cliques covering all vertices. In
this paper, we prove that for any simplicial complex or any graph, there is a
corresponding graph in class with the same well-coveredness
property. Then some necessary and sufficient conditions are presented to
recognize fast when a graph in the class is well-covered or not. To do
this characterization, we use an algebraic interpretation according to
zero-divisor elements of the edge rings of graphs.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1009.524
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity
We call a (simple) graph G codismantlable if either it has no edges or else
it has a codominated vertex x, meaning that the closed neighborhood of x
contains that of one of its neighbor, such that G-x codismantlable. We prove
that if G is well-covered and it lacks induced cycles of length four, five and
seven, than the vertex decomposability, codismantlability and
Cohen-Macaulayness for G are all equivalent. The rest deals with the
computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note
that our approach complements and unifies many of the earlier results on
bipartite, chordal and very well-covered graphs
How many matchings cover the nodes of a graph?
Given an undirected graph, are there matchings whose union covers all of
its nodes, that is, a matching--cover? A first, easy polynomial solution
from matroid union is possible, as already observed by Wang, Song and Yuan
(Mathematical Programming, 2014). However, it was not satisfactory neither from
the algorithmic viewpoint nor for proving graphic theorems, since the
corresponding matroid ignores the edges of the graph.
We prove here, simply and algorithmically: all nodes of a graph can be
covered with matchings if and only if for every stable set we have
. When , an exception occurs: this condition is not
enough to guarantee the existence of a matching--cover, that is, the
existence of a perfect matching, in this case Tutte's famous matching theorem
(J. London Math. Soc., 1947) provides the right `good' characterization. The
condition above then guarantees only that a perfect -matching exists, as
known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953).
Some results are then deduced as consequences with surprisingly simple
proofs, using only the level of difficulty of bipartite matchings. We give some
generalizations, as well as a solution for minimization if the edge-weights are
non-negative, while the edge-cardinality maximization of matching--covers
turns out to be already NP-hard.
We have arrived at this problem as the line graph special case of a model
arising for manufacturing integrated circuits with the technology called
`Directed Self Assembly'.Comment: 10 page
- …