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Let y(G) and L(G) be the domination number and independent domination number of a graph G, respectively. A graph G is called domination perfect if y(H) = L ( H ) , for every induced subgraph H of G.There are many results giving a partial characterization of domination perfect graphs. In this paper, w e present a finite induced subgraph characterization of the entire class of domination perfect graphs. The list of forbidden subgraphs in the characterization consists of 17 minimal domination imperfect graphs. Moreover, the dominating set and independent dominating set problems are shown to be both NP-complete on some classes of graphs. 0 1995 John Wiley & Sons, Inc. BASIC TERMINOLOGYAll graphs will be finite and undirected, without loops or multiple edges. If G is a graph, V ( G ) denotes the set, and IGl the number, of vertices in G. We will denote the neighborhood of a vertex x by N ( x ) . More generally, N(X) = UxtX N ( x ) for X V ( G ) . We will write x l X (x 4 ' X) to indicate that a vertex x is adjacent to all vertices (no vertex) of X C V ( G ) . For a set of vertices X, (X) denotes the subgraph of G induced by X . N ( X ) U X. In particular, if X dominates V ( G ) , then X is called a dominating set of G . An independent dominating set is a vertex subset that is both independent and dominating, or equivalently, is maximal independent. The domination Journal of Graph Theory, Vol. 20, No. 3, 375-395 (1995) If X, Y are subsets of V ( G ) , then X dominates Y if Y

Let WC( C 4 ) be the set of well-covered graphs with no cycles of length 4. The main result is that if G ∈ WC( C 4 ) then V (G) can be partitioned, using an equivalence relation, into subsets V 1 , V 2 , . . . , V k such that:; and (iii) the vector space of the well-covered weightings of G is the direct sum of the vector spaces of the well-covered weightings of the G[V i ], each of which has dimension 1.Our second result is that the problem of determining whether an edge of a graph is incident with two vertices in the same equivalence class is NP-complete. We give a forbidden co-stable subgraph characterization of graphs in WC( C 4 ). Finally, we prove that graphs in WC( C 4 ) of bounded maximum generalized degree can be recognized in polynomial time.

Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity and relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular, we recall results from the literature showing that matrix symmetrization is in fact NP-hard; furthermore, it is equivalent with the problem of recognizing whether a hypergraph can be realized as the neighborhood hypergraph of a graph. There are several variants of the latter problem corresponding to the concepts of open, closed, or mixed neighborhoods. While all these variants are NP-hard in general, one of them restricted to the bipartite graphs is known to be equivalent with graph isomorphism. Extending this result, we consider several other variants of the bipartite neighborhood recognition problem and show that they all are either polynomial-time solvable, or equivalent with graph isomorphism. Also, we study uniqueness of neighborhood realizations of hypergraphs and show that, in general, for all variants of the problem, a realization may be not unique. However, we prove uniqueness in two special cases: for the open and closed neighborhood hypergraphs of the bipartite graphs.