Tutte sets in graphs I: Maximal tutte sets and D-graphs

Tutte sets in graphs II: The complexity of finding maximum Tutte sets

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V (G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here ...

Generalized D-graphs for nonzero roots of the matching polynomial

Recently, Bauer et al. (J Graph Theory 55(4) (2007), 343–358) introduced a graph operator D(G), called the D-graph of G, which has been useful in investigating the structural aspects of maximal Tutte sets in G with a perfect matching. Among other results, they proved a characterization of maximal Tutte sets in terms of maximal independent sets in the graph D(G) and maximal extreme sets in G. Th...

Tutte Polynomials of Flower Graphs

Polynomial invariants of graphs II

Negami and Kawagoe has already defined a polynomial f(G) associated with each graph G as what discriminates graphs more finely than the polynomial f(G) definedby Negami and the Tutte polynomial. In this paper, we shall show that the polynomial f(G) includes potentially the generating function counting the independent sets and the degree sequence of a graph G, which cannot be recognized from f(G...

Bounded Treewidth Graphs – A Survey German Russian Winter School St. Petersburg, Russia

This survey gives an introduction to the class of bounded treewidth graphs, for which many NP-complete problems can be solved efficiently. The concept of a tree decomposition is explained. Subclasses of the bounded treewidth graphs are identified. Results about finding tree decompositions are summarized. Some problems which are efficiently solvable on bounded treewidth graphs are listed and alg...