Generalized activities and the tutte polynomial

Gordon, Gary; Traidi, L.

doi:10.1016/0012-365x(90)90019-e

Cited by 38 publications

(64 citation statements)

References 4 publications

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“…By Theorem 1 the restriction of 8 to G* extends to a function from G to A satisfying (5). Extend this to ( G) by linearity.…”

Section: Contraction-deletion Invariantsmentioning

confidence: 94%

“…Theorem 1. Let A be an abelian group and 8: G* Ä A a function satisfying (5). Then 8 can be extended to G so that (5) holds for ordinary edges and for bridges.…”

Section: Contraction-deletion Invariantsmentioning

confidence: 99%

“…Let A be an abelian group and 8: ( G*) Ä A a homomorphism satisfying (5). Then there is a homomorphism %: A 0 Ä A such that 8 is the restriction to ( G*) of % b Z.…”

Section: Contraction-deletion Invariantsmentioning

confidence: 99%

“…The second approach is to consider functions 8 on G imposing only (5). Let G k be the set of (isomorphism classes of) graphs with exactly k components.…”

Section: Extensionsmentioning

confidence: 99%

“…For the Tutte polynomial T(G; X, Y) there are corresponding expansions: the rankgenerating formulation, as a sum over all subgraphs, and the spanning tree expansion due to Tutte. The relationship between these expansions for T is discussed in [5]. This discussion does not apply to Z, however, as only matroid invariants are considered in [5] and Z is not a matroid invariant.…”

Section: Spanning Forest Expansionmentioning

confidence: 99%

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Contraction–Deletion Invariants for Graphs

Bollobás

Pebody²,

Riordan³

2000

Journal of Combinatorial Theory, Series B

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We consider generalizations of the Tutte polynomial on multigraphs obtained by keeping the main recurrence relation T(G)=T(GÂe)+T(G&e) for e # E(G) neither a bridge nor a loop and dropping the relations for bridges and loops. Our first aim is to find the universal invariant satisfying these conditions, from which all others may be obtained. Surprisingly, this turns out to be the universal V-function Z of Tutte (1947, Proc. Cambridge Philos. Soc. 43, 26 40) defined to obey the same relation for bridges as well. We also obtain a corresponding result for graphs with colours on the edges and describe the universal coloured V-function, which is more complicated than Z.Extending results of Tutte (1974, J. Combin. Theory Ser. B 16, 168 174) and Brylawski (1981, J. Combin. Theory Ser. B 30, 233 246), we give a simple proof that there are non-isomorphic graphs of arbitrarily high connectivity with the same Tutte polynomial and the same value of Z. We conjecture that almost all graphs are determined by their chromatic or Tutte polynomials and provide mild evidence to support this. Academic Press

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“…By Theorem 1 the restriction of 8 to G* extends to a function from G to A satisfying (5). Extend this to ( G) by linearity.…”

Section: Contraction-deletion Invariantsmentioning

confidence: 94%

“…Theorem 1. Let A be an abelian group and 8: G* Ä A a function satisfying (5). Then 8 can be extended to G so that (5) holds for ordinary edges and for bridges.…”

Section: Contraction-deletion Invariantsmentioning

confidence: 99%

“…Let A be an abelian group and 8: ( G*) Ä A a homomorphism satisfying (5). Then there is a homomorphism %: A 0 Ä A such that 8 is the restriction to ( G*) of % b Z.…”

Section: Contraction-deletion Invariantsmentioning

confidence: 99%

“…The second approach is to consider functions 8 on G imposing only (5). Let G k be the set of (isomorphism classes of) graphs with exactly k components.…”

Section: Extensionsmentioning

confidence: 99%

Section: Spanning Forest Expansionmentioning

confidence: 99%

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