UNIVERSAL GRAPHS AND INDUCED-UNIVERSAL GRAPHS

We construct graphs that contain all bounded‐degree trees on n vertices as induced subgraphs and have only cn edges for some constant c depending only on the maximum degree. In general, we consider the problem of determining the graphs, so‐called universal graphs (or induced‐universal graphs), with as few vertices and edges as possible having the property that all graphs in a specified family are contained as subgraphs (or induced subgraphs). We obtain bounds for the size of universal and induced‐universal graphs for many classes of graphs such as trees and planar graphs. These bounds are obtained by establishing relationships between the universal graphs and the induced‐universal graphs. Copyright © 1990 Wiley Periodicals, Inc., A Wiley Company