Uniquely Tree-saturated Graphs

Abstract

Let H be a graph. A graph G is uniquely H-saturated if G contains no subgraph isomorphic to H, but for every edge e in the complement of G (i.e., for each “nonedge” of G), $G+e$ contains exactly one subgraph isomorphic to H. The double star $D_{s,t}$ is the graph formed by beginning with $K_2$ and adding $s+t$ new vertices, making s of these adjacent to one endpoint of the $K_2$ and the other t adjacent to the other endpoint; $D_{s,s}$ is a balanced double star. Our main result is that the trees T for which there exist an infinite number of uniquely T-saturated graphs are precisely the balanced double stars. In addition we completely characterize uniquely tree-saturated graphs in the case where the tree is a star $K_{1,t} = D_{0, t-1}$. We show that if T is a double star, then there exists a nontrivial uniquely T-saturated graph. We conjecture that the converse holds; we verify this conjecture for all trees of order at most 6. We conclude by giving some open problems.