Generalizing D-graphs

Let ω0(G) denote the number of odd components of a graph G. The deficiency of G is defined as def(G) = maxX⊆V (G) (ω0(G−X)− |X|), and this equals the number of vertices unmatched by any maximum matching of G. A subset X ⊆ V (G) is called a Tutte set (or barrier set) of G if def(G) = ω0(G − X) − |X|, and an extreme set if def(G − X) = def(G) + |X|. Recently a graph operator, called the Dgraph D(G), was defined that has proven very useful in examining Tutte sets and extreme sets of graphs which contain a perfect matching. In this paper we give two natural and related generalizations of the D-graph operator to all simple graphs, both of which have analogues for many of the interesting and useful properties of the original.