4,420 research outputs found
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph is convex if the vertices in can be
linearly ordered such that for each vertex , the neighbors of are
consecutive in the ordering of . An induced matching of is a
matching such that no edge of connects endpoints of two different edges of
. We show that in a convex bipartite graph with vertices and
weighted edges, an induced matching of maximum total weight can be computed in
time. An unweighted convex bipartite graph has a representation of
size that records for each vertex the first and last neighbor
in the ordering of . Given such a compact representation, we compute an
induced matching of maximum cardinality in time.
In convex bipartite graphs, maximum-cardinality induced matchings are dual to
minimum chain covers. A chain cover is a covering of the edge set by chain
subgraphs, that is, subgraphs that do not contain induced matchings of more
than one edge. Given a compact representation, we compute a representation of a
minimum chain cover in time. If no compact representation is given, the
cover can be computed in time.
All of our algorithms achieve optimal running time for the respective problem
and model. Previous algorithms considered only the unweighted case, and the
best algorithm for computing a maximum-cardinality induced matching or a
minimum chain cover in a convex bipartite graph had a running time of
Maximum Matching in Turnstile Streams
We consider the unweighted bipartite maximum matching problem in the one-pass
turnstile streaming model where the input stream consists of edge insertions
and deletions. In the insertion-only model, a one-pass -approximation
streaming algorithm can be easily obtained with space , where
denotes the number of vertices of the input graph. We show that no such result
is possible if edge deletions are allowed, even if space is
granted, for every . Specifically, for every , we show that in the one-pass turnstile streaming model, in order to compute
a -approximation, space is
required for constant error randomized algorithms, and, up to logarithmic
factors, space is sufficient. Our lower bound result is
proved in the simultaneous message model of communication and may be of
independent interest
Matchings, coverings, and Castelnuovo-Mumford regularity
We show that the co-chordal cover number of a graph G gives an upper bound
for the Castelnuovo-Mumford regularity of the associated edge ideal. Several
known combinatorial upper bounds of regularity for edge ideals are then easy
consequences of covering results from graph theory, and we derive new upper
bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio
Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs
Let alpha(G) denote the maximum size of an independent set of vertices and
mu(G) be the cardinality of a maximum matching in a graph G. A matching
saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number
of vertices of G, then it is called a Konig-Egervary graph. A graph is
unicyclic if it has a unique cycle.
In 2010, Bartha conjectured that a unique perfect matching, if it exists, can
be found in O(m) time, where m is the number of edges.
In this paper we validate this conjecture for Konig-Egervary graphs and
unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm
(Karp and Spiser, 1981), which ends with an empty graph if and only if the
original graph is a Konig-Egervary graph with a unique perfect matching
obtained as an output as well.
We also show that a unicyclic non-bipartite graph G may have at most one
perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 figure
Weighted Well-Covered Claw-Free Graphs
A graph G is well-covered if all its maximal independent sets are of the same
cardinality. Assume that a weight function w is defined on its vertices. Then G
is w-well-covered if all maximal independent sets are of the same weight. For
every graph G, the set of weight functions w such that G is w-well-covered is a
vector space. Given an input claw-free graph G, we present an O(n^6)algortihm,
whose input is a claw-free graph G, and output is the vector space of weight
functions w, for which G is w-well-covered. A graph G is equimatchable if all
its maximal matchings are of the same cardinality. Assume that a weight
function w is defined on the edges of G. Then G is w-equimatchable if all its
maximal matchings are of the same weight. For every graph G, the set of weight
functions w such that G is w-equimatchable is a vector space. We present an
O(m*n^4 + n^5*log(n)) algorithm which receives an input graph G, and outputs
the vector space of weight functions w such that G is w-equimatchable.Comment: 14 pages, 1 figur
Tight upper bound on the maximum anti-forcing numbers of graphs
Let be a simple graph with a perfect matching. Deng and Zhang showed that
the maximum anti-forcing number of is no more than the cyclomatic number.
In this paper, we get a novel upper bound on the maximum anti-forcing number of
and investigate the extremal graphs. If has a perfect matching
whose anti-forcing number attains this upper bound, then we say is an
extremal graph and is a nice perfect matching. We obtain an equivalent
condition for the nice perfect matchings of and establish a one-to-one
correspondence between the nice perfect matchings and the edge-involutions of
, which are the automorphisms of order two such that and
are adjacent for every vertex . We demonstrate that all extremal
graphs can be constructed from by implementing two expansion operations,
and is extremal if and only if one factor in a Cartesian decomposition of
is extremal. As examples, we have that all perfect matchings of the
complete graph and the complete bipartite graph are nice.
Also we show that the hypercube , the folded hypercube ()
and the enhanced hypercube () have exactly ,
and nice perfect matchings respectively.Comment: 15 pages, 7 figure
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