Improved Inclusion-Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes
Klaus Dohmen, Humboldt-Universitaet zu Berlin
Abstract
Recently, Naiman and Wynn introduced the concept of an abstract tube
in order to obtain improved inclusion-exclusion identities and inequalities that involve much fewer terms than their classical counterparts.
In this paper, we introduce a particular class of abstract tubes which plays an
important role with respect to chromatic polynomials and network reliability.
The inclusion-exclusion identities and inequalities associated with this class
simultaneously generalize several well-known results such as Whitney's broken
circuit theorem, Shier's expression for the reliability of a network as an
alternating sum over chains in a semilattice and Narushima's inclusion-exclusion
identity for posets. Moreover, we show that under some restrictive assumptions
a polynomial time inclusion-exclusion algorithm can be devised, which
generalizes an important result of Provan and Ball on network reliability.
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