On the Cover Polynomial of a Digraph

Chung, Fan; Graham, Ron

doi:10.1006/jctb.1995.1055

Cited by 53 publications

(68 citation statements)

References 22 publications

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“…Graham [19], and independently in the context of rook polynomials, by I. Gessel, [37]. In [19] it is presented as an attempt to create a Tutte-like polynomial for directed graphs, and is closely related to the chromatic polynomial. There is also related work by R.P.…”

Section: Graph Polynomialsmentioning

confidence: 99%

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From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials

Makowsky

2007

Theory Comput Syst

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Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities.

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Section: Graph Polynomialsmentioning

confidence: 99%

“…Here the recursion also involves a pivot operation G ab on a graph G and two vertices a, b. In [19] such recursive definitions are given for the cover polynomial of directed graphs. B. Courcelle gave analyzed recursive definitions in depth for the interlace polynomials in [20].…”

Section: Recursive Definitionsmentioning

confidence: 99%

From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials

Makowsky

2007

Theory Comput Syst

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“…The construction of the ideal K is based on a definition of Chung and Graham [6], whose purpose was to give a combinatorial interpretation to the coefficients of the chromatic polynomial χ(n) of G when written in the basis {( n+k d )} k=0,...,d . It was shown by Chow [5] that this result can also be derived from a theorem of Stanley's concerning his chromatic symmetric function [9].…”

Section: Introductionmentioning

confidence: 99%

“…Then the path lengths (k) associated to π are given by 5 2 3 6 4 1 7 0 0 1 0 2 1 1 so π has cuts 0, 2, 4 and 6 and thus G-sequence {2, 5}, {3, 6}, {1, 4}, {7}. The following theorem is stated (in a different, but equivalent, form) in [6], where it is claimed that it follows from the work of Brenti in [2]. Also, in [4] it is shown that the theorem follows from certain properties of the chromatic symmetric function of Stanley [9].…”

Section: Introductionmentioning

confidence: 99%

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Steingrı́msson¹

2001

Journal of Algebraic Combinatorics

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Abstract. Let G be a simple graph on d vertices. We define a monomial ideal K in the Stanley-Reisner ring A of the order complex of the Boolean algebra on d atoms. The monomials in K are in one-to-one correspondence with the proper colorings of G. In particular, the Hilbert polynomial of K equals the chromatic polynomial of G.The ideal K is generated by square-free monomials, so A/K is the Stanley-Reisner ring of a simplicial complex C. The h-vector of C is a certain transformation of the tail T (n) = n d − χ(n) of the chromatic polynomial χ of G. The combinatorial structure of the complex C is described explicitly and it is shown that the Euler characteristic of Cequals the number of acyclic orientations of G.

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“…Besides their recursive definition, like in the case of the Tutte polynomial and its close relatives, cf. [5,6,7], they usually also have an equivalent (up to some transformation) static definition as some kind of generating function. The matching polynomial e.g.…”

Section: Msol-polynomialsmentioning

confidence: 99%

Linear Recurrence Relations for Graph Polynomials

Fischer

Makowsky

Pillars of Computer Science

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For Boaz (Boris) Abramovich Trakhtenbrot on the occasion of his 85th birthday. Abstract.A sequence of graphs Gn is iteratively constructible if it can be built from an initial labeled graph by means of a repeated fixed succession of elementary operations involving addition of vertices and edges, deletion of edges, and relabelings. Let Gn be a iteratively constructible sequence of graphs. In a recent paper, [27], M. Noy and A. Ribò have proven linear recurrences with polynomial coefficients for the Tutte polynomials T (Gi, x, y) = T (Gi), i.e. T (Gn+r) = p1(x, y)T (Gn+r−1) + . . . + pr(x, y)T (Gn).We show that such linear recurrences hold much more generally for a wide class of graph polynomials (also of labeled or signed graphs), namely they hold for all the extended MSOL-definable graph polynomials. These include most graph and knot polynomials studied in the literature.

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On the Cover Polynomial of a Digraph

Cited by 53 publications

References 22 publications

From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials

From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials

Untitled

Linear Recurrence Relations for Graph Polynomials

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