RECTANGULARLY DUALIZABLE GRAPHS: AREA-UNIVERSALITY

A Dualizable Representation for Imbedded Graphs

A new representation is described for graphs imbedded in pseudomanifoldr. The graphs may have loops and parallel edges, and they may be disconnected. It is shown that our representation has a well-dejined dual that naturally extends the definition of surface duality. This representation is easily transformed to a data structure with a set of operators for modifying a graph, a set of query opera...

An Algebra That Is Dualizable but Not Fully Dualizable

We give an example of a finite algebra which is dualizable but not fully dualizable in the sense of natural duality theory.

Z8 Is Not Dualizable

Almost-spanning universality in random graphs

A graph G is said to be H(n,∆)-universal if it contains every graph on at most n vertices with maximum degree at most ∆. It is known that for any ε > 0 and any natural number ∆ there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely H((1− ε)n,∆)universal for p ≥ c(log n/n). Bypassing this natural boundary, we show that for ∆ ≥ 3 the same conclusion holds when p n− ...

On the Universality of Small-World Graphs

The clustering of a graph is defined as the fraction of existing edges between the neighbors of a vertex and the maximum number of edges that could possibly exist among the neighbors, averaged over all the vertices of the graph. Unlike other important graph parameters the clustering is a local measure. A small-world graph is a graph which has a relatively low diameter (with respect to a regular...