321 research outputs found
Enumeration of Matchings in the Incidence Graphs of Complete and Complete Bipartite Graphs
If G = (V, E) is a graph, the incidence graphI(G) is the graph with vertices I βͺ E and an edge joining v β V and e β E when and only when v is incident with e in G. For G equal to Kn (the complete graph on n vertices) or Kn,n (the complete bipartite graph on n + n vertices), we enumerate the matchings (sets of edges, no two having a vertex in common) in I(G), both exactly (in terms of generating functions) and asymptotically. We also enumerate the equivalence classes of matchings (where two matchings are considered equivalent if there is an automorphism of G that induces an automorphism of I(G) that takes one to the other)
Optimum matchings in weighted bipartite graphs
Given an integer weighted bipartite graph we consider the problems of finding all the edges that occur in
some minimum weight matching of maximum cardinality and enumerating all the
minimum weight perfect matchings. Moreover, we construct a subgraph of
which depends on an -optimal solution of the dual linear program
associated to the assignment problem on that allows us to reduced
this problems to their unweighed variants on . For instance, when
has a perfect matching and we have an -optimal solution of the dual
linear program associated to the assignment problem on , we solve the
problem of finding all the edges that occur in some minimum weight perfect
matching in linear time on the number of edges. Therefore, starting from
scratch we get an algorithm that solves this problem in time
, where , , and .Comment: 11 page
JGraphT -- A Java library for graph data structures and algorithms
Mathematical software and graph-theoretical algorithmic packages to
efficiently model, analyze and query graphs are crucial in an era where
large-scale spatial, societal and economic network data are abundantly
available. One such package is JGraphT, a programming library which contains
very efficient and generic graph data-structures along with a large collection
of state-of-the-art algorithms. The library is written in Java with stability,
interoperability and performance in mind. A distinctive feature of this library
is the ability to model vertices and edges as arbitrary objects, thereby
permitting natural representations of many common networks including
transportation, social and biological networks. Besides classic graph
algorithms such as shortest-paths and spanning-tree algorithms, the library
contains numerous advanced algorithms: graph and subgraph isomorphism; matching
and flow problems; approximation algorithms for NP-hard problems such as
independent set and TSP; and several more exotic algorithms such as Berge graph
detection. Due to its versatility and generic design, JGraphT is currently used
in large-scale commercial, non-commercial and academic research projects. In
this work we describe in detail the design and underlying structure of the
library, and discuss its most important features and algorithms. A
computational study is conducted to evaluate the performance of JGraphT versus
a number of similar libraries. Experiments on a large number of graphs over a
variety of popular algorithms show that JGraphT is highly competitive with
other established libraries such as NetworkX or the BGL.Comment: Major Revisio
Symmetries of plane partitions and the permanent-determinant method
In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives
formulas for the number of plane partitions in each of ten symmetry classes.
This paper together with results by Andrews [J. Combin. Theory Ser. A 66
(1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the
project of proving all ten formulas.
We enumerate cyclically symmetric, self-complementary plane partitions. We
first convert plane partitions to tilings of a hexagon in the plane by
rhombuses, or equivalently to matchings in a certain planar graph. We can then
use the permanent-determinant method or a variant, the Hafnian-Pfaffian method,
to obtain the answer as the determinant or Pfaffian of a matrix in each of the
ten cases. We row-reduce the resulting matrix in the case under consideration
to prove the formula. A similar row-reduction process can be carried out in
many of the other cases, and we analyze three other symmetry classes of plane
partitions for comparison
- β¦