### Violating Helly or Leray property

The Helly property is definitively needed.
For instance, convex compact subsets of **R**^{d}
have the Leray property, but not the Helly property.
Surprisingly all graphs are intersection graphs of
convex compact subsets of **R**^{3}, and all
octahedrons (but not all graphs) are intersection graphs of
convex compact subsets of **R**^{2}, compare
[W67].

What happens if only relaxed versions of the Leray-property
are valid: Either we have *0=
*_{k}=
_{k+1} =...
for every nonempty intersection of *S*_{x}s,
for some fixed integer *k*, or we request
*k
*_{0},
_{1},
_{2},...
for every nonempty intersection of *S*_{x}s,
again for some fixed *k*? |

Circular arcs of a cycle are an example where both the
Helly property and the Leray property may be violated, but
only slightly.
The Helly property is violated if there are three arcs
covering the whole circle,
and the Leray property is violated if there are two
arcs covering the whole circle.
It quickly follows from the
Theorem
on intersection graphs of dual hypergraphs that
_{1}(G^)
=0 or 1
for every circular-arc graph *G*, depending on whether
the arcs cover the whole circle and no three cover the whole
circle.

What can be said on the other Betti numbers of the complexes
of circular-arc graphs? |

Back to the
start page for intersection graphs.

Erich Prisner

made on January 12, 1999,
last changed on February 5, 1999.