Universal graphs with a forbidden subtree

The systematic investigation of countable universal graphs with “forbidden” subgraphs was initiated in [13], followed by [12]. If C is a finite connected graph, then a graph G is C-free if it contains no subgraph isomorphic to C. A countable C-free graph G is weakly universal if every countable C-free graph is isomorphic to a subgraph of G, and strongly universal if every such graph is isomorphic to an induced subgraph of G. Such universal graphs, in either sense, are rare. Graph theorists tend to use the term “universal” in the weak sense, while model theorists tend to use it in the strong sense. We will use the term in the graph-theoretical sense here: “universal” means “weakly universal”, though we sometimes include the adverb for emphasis. Similarly, while model theorists may sometimes use the term “subgraph” for “induced subgraph,” we avoid such usage here. We deal here with the problem of determining the finite connected constraint graphs C for which there is a countable universal C-free graph. We introduce a new inductive method and use it to settle the case in which C is a tree, confirming a long-standing conjecture of Tallgren. The existence of such a countable universal graph says something about the class of all finite