A Tutte Polynomial for Partially Ordered Sets

Gordon, Gary

doi:10.1006/jctb.1993.1060

Cited by 12 publications

(11 citation statements)

References 10 publications

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“…Some of these interpretations are closely related to matroidal or graphical properties of β. This lends support to our view that the Tutte and characteristic polynomials studied in [11,14,15,16,17,18,19] are (in some sense) also the 'right' generalizations to greedoids.…”

Section: Introductionsupporting

confidence: 69%

A $\beta$ Invariant for Greedoids and Antimatroids

Gordon

1997

Electron. J. Combin.

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We extend Crapo's $\beta $ invariant from matroids to greedoids, concentrating especially on antimatroids. Several familiar expansions for $\beta (G)$ have greedoid analogs. We give combinatorial interpretations for $\beta (G)$ for simplicial shelling antimatroids associated with chordal graphs. When $G$ is this antimatroid and $b(G)$ is the number of blocks of the chordal graph $G$, we prove $\beta (G)=1-b(G)$.

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Section: Introductionsupporting

confidence: 69%

A $\beta$ Invariant for Greedoids and Antimatroids

Gordon

1997

Electron. J. Combin.

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“…At present there does not seem to be a satisfactory definition of a Tutte polynomial for oriented matroids. We remark that in Gordon [Go93], a Tutte-like polynomial is defined for posets. Also, Gessel [Ge89] has introduced a two variable version of the rook polynomial which takes into account the cycle structure of the rook placements.…”

Section: Discussionmentioning

confidence: 99%

On the Cover Polynomial of a Digraph

Chung

Graham

1995

Journal of Combinatorial Theory, Series B

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“…The probabilistic approach taken here should be applicable to many of the combinatorial structures considered here. This could include a reliability theory for: * Partially ordered sets [9,10] * Rooted directed graphs [20] * Convex point sets [1,8] * Simplicial shelling in a chordal graph [11] See [19] for more examples of greedoids.…”

Section: Applications To Other Antimatroids and Greedoidsmentioning

confidence: 99%

When bad things happen to good trees*

Aivaliotis

Gordon

Graveman

2001

Journal of Graph Theory

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When the edges in a tree or rooted tree fail with a certain ®xed probability, the (greedoid) rank may drop. We compute the expected rank as a polynomial in p and as a real number under the assumption of uniform distribution. We obtain several different expressions for this expected rank polynomial for both trees and rooted trees, one of which is especially simple in each case. We also prove two extremal theorems that determine both the largest and smallest values for the expected rank of a (rooted or unrooted) tree, and precisely when these extreme bounds are achieved. We conclude with directions for further study. ß

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A Tutte Polynomial for Partially Ordered Sets

Cited by 12 publications

References 10 publications

A $\beta$ Invariant for Greedoids and Antimatroids

A $\beta$ Invariant for Greedoids and Antimatroids

On the Cover Polynomial of a Digraph

When bad things happen to good trees*

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