Stable Matchings in Trees
نویسندگان
چکیده
The maximum stable matching problem (Max-SMP) and the minimum stable matching problem (Min-SMP) have been known to be NP-hard for subcubic bipartite graphs, while Max-SMP can be solved in polynomal time for a bipartite graph G with a bipartition (X,Y ) such that degG(v) ≤ 2 for any v ∈ X. This paper shows that both Max-SMP and Min-SMP can be solved in linear time for trees. This is the first polynomially solvable case for Min-SMP, as far as the authors know. We also consider some extensions to the case when G is a general/bipartite graph with edge weights.
منابع مشابه
The number of maximum matchings in a tree
We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) [Formula: see text] of a tree T of given order. While the trees that attain the lower bound are easily characterised, the trees with the largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of ...
متن کامل0 Trees and Matchings Richard
In this article, Temperley’s bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spann...
متن کاملOrdering Trees with Perfect Matchings by Their Wiener Indices
The Wiener index of a connected graph is the sum of all pairwise distances of vertices of the graph. In this paper, we consider the Wiener indices of trees with perfect matchings, characterizing the eight trees with smallest Wiener indices among all trees of order 2 ( 11) m m with perfect matchings.
متن کاملOn the Total Number of Matchings of Trees with Prescribed Diameter
Let G = (V (G), E(G)) be a graph. An m−matchings of G is a set of edges of size m in which any two edges are mutually independent. Denote by z(G,m) the number of m−matchings of G. Let z(G) be the total number of matchings in G, namely z(G) = bn 2 c ∑ m=1 z(G,m). It’s well-known that z(G) are also named as Hosoya index. Let Tn,d be the set of trees of on n vertices with diameter d. In this paper...
متن کاملRandom Walks on Trees and Matchings
We give sharp rates of convergence for a natural Markov chain on the space of phylogenetic trees and dually for the natural random walk on the set of perfect matchings in the complete graph on 2n vertices. Roughly, the results show that 1 2 n log n steps are necessary and suffice to achieve randomness. The proof depends on the representation theory of the symmetric group and a bijection between...
متن کامل