%0 Journal Article %T Do Almost All Trees Have No Perfect Dominating Set? %A Bill Quan Yue %J Open Journal of Discrete Mathematics %P 1-13 %@ 2161-7643 %D 2018 %I Scientific Research Publishing %R 10.4236/ojdm.2018.81001 %X A graph G is said to have a perfect dominating set S if S is a set of vertices of G and for each vertex v of G, either v is in S and v is adjacent to no other vertex in S, or v is not in S but is adjacent to precisely one vertex of S. A graph G may have none, one or more than one perfect dominating sets. The problem of determining if a graph has a perfect dominating set is NP-complete. The problem of calculating the probability of an arbitrary graph having a perfect dominating set seems also difficult. In 1994 Yue [1]