21,152 research outputs found
Generalization of matching extensions in graphs (II)
Proposed as a general framework, Liu and Yu(Discrete Math. 231 (2001)
311-320) introduced -graphs to unify the concepts of deficiency of
matchings, -factor-criticality and -extendability. Let be a graph and
let and be non-negative integers such that and
is even. If when deleting any vertices from , the remaining
subgraph of contains a -matching and each such - matching can be
extended to a defect- matching in , then is called an
-graph. In \cite{Liu}, the recursive relations for distinct parameters
and were presented and the impact of adding or deleting an edge also
was discussed for the case . In this paper, we continue the study begun
in \cite{Liu} and obtain new recursive results for -graphs in the
general case .Comment: 12 page
Generalization of matching extensions in graphs—combinatorial interpretation of orthogonal and q-orthogonal polynomials
AbstractIn this paper, we present generalization of matching extensions in graphs and we derive combinatorial interpretation of wide classes of orthogonal and q-orthogonal polynomials. Specifically, we assign general weights to complete graphs, cycles and chains or paths defining matching extensions in these graphs. The generalized matching polynomials of these graphs have recurrences defining various orthogonal polynomials—including classical and non-classical ones—as well as q-orthogonal polynomials. The Hermite, Gegenbauer, Legendre, Chebychev of the first and second kind, Jacobi and Pollaczek orthogonal polynomials and the continuous q-Hermite, Big q-Jacobi, Little q-Jacobi, Al Salam and alternative q-Charlier q-orthogonal polynomials appeared as applications of this study
Submodular Maximization Meets Streaming: Matchings, Matroids, and More
We study the problem of finding a maximum matching in a graph given by an
input stream listing its edges in some arbitrary order, where the quantity to
be maximized is given by a monotone submodular function on subsets of edges.
This problem, which we call maximum submodular-function matching (MSM), is a
natural generalization of maximum weight matching (MWM), which is in turn a
generalization of maximum cardinality matching (MCM). We give two incomparable
algorithms for this problem with space usage falling in the semi-streaming
range---they store only edges, using working memory---that
achieve approximation ratios of in a single pass and in
passes respectively. The operations of these algorithms
mimic those of Zelke's and McGregor's respective algorithms for MWM; the
novelty lies in the analysis for the MSM setting. In fact we identify a general
framework for MWM algorithms that allows this kind of adaptation to the broader
setting of MSM.
In the sequel, we give generalizations of these results where the
maximization is over "independent sets" in a very general sense. This
generalization captures hypermatchings in hypergraphs as well as independence
in the intersection of multiple matroids.Comment: 18 page
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