categories: Re: graph classifiers

[John Stell <j.g.stell@cs.keele.ac.uk>]

To clarify what I meant by the graph Delta in my earlier message,
we can proceed via hypergraphs. In what follows all graphs are
undirected, so Omega has two nodes and four edges. Define a hypergraph
to be a function h : E -> PN, where E and N are sets of edges and nodes
and PN is the powerset of N. Each hypergraph has dual h* : N -> PE
where h* n = {e \in E | n \in h e}. 

A morphism of hypergraphs is a pair of functions phi_N : N_1 -> N_2
and phi_E : E_1 -> E_2 such that P phi_N h_1 subseteq h_2 phi_E.
If H and K are graphs a hypergraph morphism from H to K may not
be a graph morphism since a loop can be mapped to an edge which is not
a loop. However, given any hypergraph h we can define a graph having
the same nodes as h but with an edge joining node x to y for every
edge in h incident with both x and y. If this graph is denoted
by G(h), hypergraph morphisms from a graph K to a hypergraph h
correspond to graph morphisms from K to G(h). The graph I called 
Delta before is G(Omega*) it has four nodes and nine edges.

To explain the interpretation of Omega*, let's denote the nodes of
Omega by 0 and 1 and the four edges by {1}1, {1}0, {0,1}0, {0}0.
Given a subgraph gamma : H -> Omega, the nodes and edges have the
following interpretations.
  0      nodes not in the subgraph
  1      nodes in the subgraph
  {1}1   edges in the subgraph
  {1}0   edges not in the subgraph but with both end nodes in 
  {0,1}0 edges not in the subgraph but with one end in and one out
  {0}0   edges not in the subgraph with both end nodes out.

Now give Omega* the following interpretation
  0      edges in the subgraph
  1      edges not in the subgraph
  {1}1   nodes not in the subgraph
  {1}0   nodes in the subgraph which are ends of a non-empty set of
         edges all of which are out of the subgraph.
  {0,1}0 nodes in the subgraph which are ends of some edges in the 
         subgraph and ends of some edges which are out of the subgraph.
  {0}0   nodes in the subgraph having all their incident edges in the 
         subgraph, or having no incident edges.

Given any subgraph gamma : H -> Omega, we get neg gamma from
the endomorphism of Omega switching 0 and 1 and taking {0}0
to {1}1. Using the above interpretation of Omega* we can construct a
hypergraph morphism gamma! : H -> Omega*. I don't have a neat
construction for gamma! except via the above interpretation of Omega*.
Now the endomorphism neg : Omega -> Omega dualizes to
neg* : Omega* -> Omega* and we compose this with gamma! to
get a hypergraph morphism from H to Omega* which represents suppl gamma.


John Stell