In this note we study a certain graph polynomial arising from a special recursion. This recursion is a member of a family of four recursions where the other three recursions belong to the chromatic polynomial, the modified matching polynomial, and the adjoint polynomial, respectively. The four polynomials share many common properties, for instance all of them are of exponential type, i. e., the...

Recently, Tittmann et al. introduced the subgraph component polynomial and showed that its power for distinguishing graphs is quite different from the power of other graph polynomials that appear in the literature such as the matching polynomial, the Tutte polynomial, the characteristic polynomial, the chromatic polynomial, etc. The subgraph component polynomial enumerates vertex induced subgra...

Matching 2-lattice polyhedra are a special class of lattice polyhedra that include network 3ow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, etc. In this paper we develop a polynomial-time extreme point algorithm for #nding a maximum cardinality vector in a matching 2-lattice polyhedron. c © 2001 Elsevier Science B.V. All rights reserved.

Recently, Tittmann et al. introduced the subgraph component polynomial and showed that its power for distinguishing graphs is quite different from the power of other graph polynomials that appear in the literature such as the matching polynomial, the Tutte polynomial, the characteristic polynomial, the chromatic polynomial, etc. The subgraph component polynomial enumerates vertex induced subgra...

In two-sided matching markets, the concept of stability proposed by Gale and Shapley (1962) is one of the most important solution concepts. In this paper, we consider a problem related to the stability of a matching in a two-sided matching market with indifferences (i.e., ties). The introduction of ties into preference lists dramatically changes the properties of stable matchings. For example, ...

For a connected graph G=(V,E), matching M⊆E is cut of G if G−M disconnected. It known that for an integer d, the corresponding decision problem Matching Cut polynomial-time solvable graphs diameter at most d d≤2 and NP-complete d≥3. We prove same dichotomy bounded radius. H, H-free it does not contain H as induced subgraph. As consequence our result, we can solve in polynomial time P6-free grap...

We show that feasible elimination procedures (Peleg, 1978) can be used to select k from m alternatives. An important advantage of this method is the core property: no coalition can guarantee an outcome that is preferred by all its members. We also provide an axiomatic characterization for the case k = 1, using the conditions of anonymity, Maskin monotonicity, and independent blocking. Finally, ...

School of Mathematical and Statistical Sciences, Arizona State University Abstract. Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings on [2n] := {1, 2, . . . , 2n} using the cyclic group C2n generated by a cyclic shift of order 2n and the q-Cata...

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre poly...

We study the zeros of two families of polynomials related to rook theory and matchings in graphs. One of these families is based on the cover polynomial of a digraph introduced by Chung and Graham [ChGr]. Another involves a version of the \hit polynomial" of rook theory, but which applies to weighted matchings in (non-bipartite) graphs. For both of these families we prove a result which is anal...