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

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 of G. 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. In this article, we study the structural aspects of maximal Tutte sets in a graph G. Towards this end, we introduce a related graph D(G). We first show that the maximal Tutte sets in G are precisely the maximal independent sets in its D-graph D(G), and then continue with the study of D-graphs in their own right, and of iterated D-graphs. We show that G is isomorphic to a spanning subgraph of D(G), and characterize the graphs for which G≅D(G) and for which D(G)≅D2(G). Surprisingly, it turns out that for every graph G with a perfect matching, D3(G)≅D2(G). Finally, we characterize bipartite D-graphs and comment on the problem of characterizing D-graphs in general. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 343–358, 2007