On Universal Graphs of Minor Closed Families

For a family F of graphs, a graph U is said to be F-universal if every graph of F is a subgraph of U. Similarly, a graph is said to be F-induced-universal if every graph of F is an induced subgraph of U. In this paper, we give constructive proofs of new upper bounds for size and order of such minimal graphs for the family of graphs with no K-minor and more particularly on graphs of bounded treewidth.