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1999 Improved Inclusion-Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes
Klaus Dohmen
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Electron. J. Probab. 4: 1-12 (1999). DOI: 10.1214/EJP.v4-42


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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Klaus Dohmen. "Improved Inclusion-Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes." Electron. J. Probab. 4 1 - 12, 1999.


Accepted: 26 March 1999; Published: 1999
First available in Project Euclid: 4 March 2016

zbMATH: 0920.05008
MathSciNet: MR1684161
Digital Object Identifier: 10.1214/EJP.v4-42

Primary: 05A19
Secondary: 05A20 , 05C15 , 60C05 , 68M15 , 90B12 , 90B25

Keywords: abstract simplicial complex , abstract tube , Bonferroni inequalities , broken circuit complex , chain , chromatic polynomial , dynamic programming , Graph coloring , inclusion-exclusion , Network reliability , partial order , sieve formula

Vol.4 • 1999
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