Unique domination and domination perfect graphs

We review a characterization of domination perfect graphs in terms of forbidden induced subgraphs obtained by Zverovich and Zverovich [12] using a computer code. Then we apply it to a problem of unique domination in graphs recently proposed by Fischermann and Volkmann. 1 Domination perfect graphs Let G be a graph. A set D ⊆ V (G) is a dominating set of G if each vertex of G either belongs to D or it is adjacent to a vertex of D. A set S ⊆ V (G) is stable [or independent] if it induces an edgeless subgraph G(S). An independent dominating set is vertex subset that is both stable and dominating, or equivalently, is maximal stable. The domination number γ(G) is the minimum cardinality taken over all dominating sets of G, and the independent domination number ι(G) is the minimum cardinality taken over all maximal stable sets of vertices of G. Let X ⊆ V (G). For a vertex x ∈ X, the set PN(x,X) = N [x]\N [X\{x}] is called the private neighborhood of x [with respect to X]. A dominating set of a graph G is minimal if it does not contain another dominating set of G. It is wellknown and easy to see that a dominating set D is minimal if and only if PN(x,D) 6= ∅ for each x ∈ D [Berge’s lemma]. Definition 1 (Sumner and Moore [8]). A graph G is called domination perfect or γ-perfect if γ(H) = ι(H) for every induced subgraph H of G. A graph G is called minimal domination imperfect if G is not domination perfect and γ(H) = ι(H) for every proper induced subgraph H of G. We shortly describe a history of the forbidden induced subgraph characterization of domination perfect graphs [12]. First Allan and Laskar noted thatK1,3-free graphs are domination perfect. A more particular results that line graphs and middle graphs are domination perfect was obtained by Gupta [see Theorem 10.5 in [5]]. Theorem 1 (Sumner and Moore [8]). A graph G is domination perfect if and only if γ(H) = ι(H) for every induced subgraph H of G with γ(H) = 2. Let A = {H : |V (H)| ≤ 8, γ(H) = 2, and ι(H) > 2}. Sumner and Moore [8] proved a series of results on domination perfect graphs: • A chordal graph is domination perfect if and only if it is G1-free (Figure 3). • If G is an A-free graph and also G does not contain an induced copy of the graph S in Figure 1= , then G is domination perfect. • A planar graph is domination perfect if and only if it is A-free. – 2 – uu u uu u uu PPPPPPPPP @ @ @ @ @ @ @ @ @