Let G be a graph on n vertices and m edges and D(G, x) the domination polynomial of G. In this paper we completely characterize the values of n and m for which optimal graphs exist for domination polynomials. We also show that there does not always exist least optimal graphs for the domination polynomial. Applications to network reliability are highlighted.

中文翻译：

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最优控制多项式

令G为n个顶点和m个边上的图，D（G，  x）为G的支配多项式。在本文中，我们完全表征了存在于控制多项式最优图的n和m值。我们还表明，对于控制多项式，并不总是存在最小最优图。强调了网络可靠性的应用。