We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce the class of (hyper)graph polynomials definable in second order logic, and outline a research progr...

We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we establish common generalizations of the main results on those two problems--polynomial-time solvability, rain-max theorems, and totally dual integral polyhedral descriptions. New applications of these results include a strongly polynomial separation algorithm fo...

Let G = (V,E) be a finite undirected graph without loops and multiple edges. An edge set E ⊆ E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of E. In particular, this means that E is an induced matching, and every edge not in E shares exactly one vertex with an edge in E. Clearly, not every graph has a d.i.m. The Dominating Induced Matching...

The 0/1 primal separation problem is: Given an extreme point x̄ of a 0/1 polytope P and some point x , find an inequality which is tight at x̄, violated by x and valid for P or assert that no such inequality exists. It is known that this separation variant can be reduced to the standard separation problem for P. We show that 0/1 optimization and 0/1 primal separation are polynomial time equivalen...

Let H be a (k + 1)-uniform, D-regular hypergraph on n vertices and U(H) be the minimum number of vertices left uncovered by a matching in H. Cj(H), the j-codegree of H, is the maximum number of edges sharing a set of j vertices in common. We prove a general upper bound on U(H), based on the codegree sequence C2(H), C3(H) . . . . Our bound improves and generalizes many results on the topic, incl...

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS). We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the general...

Many polynomials have been de'ned associated to graphs, like the characteristic, matchings, chromatic and Tutte polynomials. Besides their intrinsic interest, they encode useful combinatorial information about the given graph. It is natural then to ask to what extent any of these polynomials determines a graph and, in particular, whether one can 'nd graphs that can be uniquely determined by a g...

We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland’s Lower Matching Conjecture and Schrijver’s theorem. It extends the recent work of Csikvári done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of 2-li...

We study the total weight of weighted matchings in segment graphs, which is related to a question concerning generalized Chebyshev polynomials introduced by Vauchassade de Chaumont and Viennot and, more recently, investigated by Kim and Zeng. We prove that weighted matchings with sufficiently large node-weight cannot have equal total weight.

We derive a polynomial time algorithm to compute a stable solution in a mixed matching market from an auction procedure as presented by Eriksson and Karlander [5]. As a special case we derive an O(nm) algorithm for bipartite matching that does not seem to have appeared in the literature yet.