Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs

Let alpha(G) denote the maximum size of an independent set of vertices and mu(G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number of vertices of G, then it is called a Konig-Egervary graph. A graph is unicyclic if it has a unique cycle. In 2010, Bartha conjectured that a unique perfect matching, if it exists, can be found in O(m) time, where m is the number of edges. In this paper we validate this conjecture for Konig-Egervary graphs and unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm (Karp and Spiser, 1981), which ends with an empty graph if and only if the original graph is a Konig-Egervary graph with a unique perfect matching obtained as an output as well. We also show that a unicyclic non-bipartite graph G may have at most one perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 figure

Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a greedoid if and only if G has a unique perfect matching.Comment: 12 pages, 12 figure

Dependencies among dependencies in matroids

In 1971, Rota introduced the concept of derived matroids to investigate “de-pendencies among dependencies” in matroids. In this paper, we study the derived matroid δM of an F-representation of a matroid M. The matroid δM has a naturally associated F-representation, so we can define a sequence δM, δ2M,. The main result classifies such derived sequences of matroids into three types: finite, cyclic, and divergent. For the first two types, we obtain complete characterizations and thereby resolve some of the questions that Longyear posed in 1980 for binary matroids. For the last type, the divergence is estimated by the coranks of the matroids in the derived sequence

Arbres minimaux d'un graphe preordonne

AbstractWe study the minimal spanning trees of a connected graph G = (X,U) where U is partially preordered (or quasi-ordered). We characterize several kinds of optimal spanning trees and give conditions for existence of strongly optimal trees. Generalizations to bases of matroids (binary matroïds in part 2) are immediate. Sone of our results are given in terms of Krugdahl's dependence graphs. They imply previous results of Rosenstiehl and Gale in the case of linear orders or preorders.RésuméOn étudie les arbres minimaux d'un graphe connexe G = (X,U), où U est partiellement préordonné. On caractérise diverses sortes d'arbes optimaux et on donne des conditions d'existence des arbres fortement optimaux. Des généralisations aux bases de matroides (matroides binaires dans la partie 2) sont immédiates. Certains de nos résultats utilisent les graphes de déviance de Krogdahl. Its impliquent des résultats de Rosenstiehl et de Gale dans les cas dependances et de préorders totaux

Matroids and linking systems

AbstractWith the help of the concept of a linking system, theorems relating matroids with bipartite graphs and directed graphs are deduced. In this way natural generalizations of theorems of Edmonds & Fulkerson, Perfect, Pym. Rado, Brualdi and Mason are obtained. Furthermore some other properties of these linking systems are investigated

The Minimum Cost Query Problem on Matroids with Uncertainty Areas

We study the minimum weight basis problem on matroid when elements\u27 weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements. Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base

Market Pricing for Matroid Rank Valuations

In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable to achieve optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a simple partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by D\"uetting and V\'egh in 2017. We further formalize a weighted variant of the conjecture of D\"uetting and V\'egh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M${}^\natural$-concave functions, or equivalently, gross substitutes functions