We give a bijection between partially directed paths in the symmetric wedge y = ±x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of Prellberg et al. that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y = ±x with k north steps is equa...

In this paper, we study the distribution of the number of consecutive pattern matches of the five up-down permutations of length four, 1324, 2314, 2413, 1432, and 3412, in the set of up-down permutations. We show that for any such τ , the generating function for the distribution of the number of consecutive pattern matches of τ in the set of up-down permutations can be expressed in terms of wha...

For a graph G let L(G) and l(G) denote the size of the largest and smallest maximum matching of a graph obtained from G by removing a maximum matching of G. We show that L(G) ≤ 2l(G), and L(G) ≤ 3 2 l(G) provided that G contains a perfect matching. We also characterize the class of graphs for with L(G) = 2l(G). Our characterization implies the existence of a polynomial algorithm for testing the...

Let [Formula: see text] be an instance of the stable marriage problem in which every vertex ranks its neighbors a strict order preference. A matching is popular if does not lose head-to-head election against any matching. Popular matchings generalize matchings. Unfortunately, when there are edge costs, to find or even approximate up factor minimum cost NP-hard. min-cost Our goal efficiently com...

Given an undirected graph G = (V,E) and a delta-matroid (V,F), the delta-matroid matching problem is to find a maximum cardinality matching M such that the set of the end vertices of M belongs to F . This problem is a natural generalization of the matroid matching problem to delta-matroids, and thus it cannot be solved in polynomial time in general. This paper introduces a class of the delta-ma...

For an arbitrary tree we investigate the problems of constructing a maximum matching which minimizes or maximizes the cardinality of a maximum matching of the graph obtained from original one by its removal and present corresponding polynomial algorithms.

Suppose we have a well-solved optimization problem, such as minimum spanning tree, maximum cut in planar graphs, minimum weight perfect matching, or maximum weight independent set in a bipartite graph. How hard is it to determine whether there exists a solution with a given weight ? Papadimitriou and Yannakakis showed in [PAPADIMITRIOU 82] that these so-called exact versions of the above optimi...

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...