Universal Graphs for Two Graph Properties

The graphs we consider are all countable. A graph U is universal in a given set P of graphs if every graph in P is an induced subgraph of U and U is an element of P. In this paper we show the existence of a universal graph in the set of all countable graphs with block order bounded by a fixed positive integer. We also investigate some classes of interval graphs and work towards finding universal graphs for them. The sets of graphs we consider are all examples of induced-hereditary graph properties.