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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

The bondage number b(G) of a graph G is the smallest number of edges of G whose removal from G results in a graph having the domination number larger than that of G. We show that, for a graph G having the maximum vertex degree $\Delta(G)$ and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, $b(G)\le \min\{\Delta(G)+h+2, \Delta(G)+k+1\}$. This generalizes known upper bounds for planar and toroidal graphs.Comment: 10 pages; Updated version (April 2011); Presented at the 7th ECCC, Wolfville (Nova Scotia, Canada), May 4-6, 2011, and the 23rd BCC, Exeter (England, UK), July 3-8, 201

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