Bicycle Dimension and Special Points of the Tutte Polynomial

Vertigan, Dirk

doi:10.1006/jctb.1998.1860

Cited by 25 publications

(44 citation statements)

References 17 publications

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“…Vertigan announced that this problem is #P-complete for any fixed field F in [14], but his result remains unpublished.…”

Section: Counting Basesmentioning

confidence: 99%

“…Vertigan [14] shows that if V is a linear subspace of F E and |F| is one of 2, 3, 4, then d(V ) depends only on M(V ).…”

Section: The Bicycle Dimensionmentioning

confidence: 99%

“…Extending the work of Jaeger, Vertigan, and Welsh [7], Vertigan investigates the complexity of τ 0 (M F , x, y), the problem of evaluating T M (x, y) given x, y and a matroid M that is linearly represented over F [14]. Let ι denote the imaginary unit, let ω := e (2) |F| = 3, and (x, y) is one of (ω, ω 2 ), (ω 2 , ω); or (3) |F| = 4, and (x, y) is (−1, −1).…”

Section: Introductionmentioning

confidence: 99%

“…(1) If (x, y) = (0, 0) or if (x − 1)(y − 1) = 1, it is trivial to compute T M (x, y) for any matroid M; (2) for binary matroids M, |T M (−ι, ι)| was determined by Vertigan [14]; below, we derive an explicit expression for T M (−ι, ι); (3) for ternary matroids M, Jaeger [8] has determined |T M (ω, ω 2 )|, and Gioan and Las Vergnas [3] found T M (ω, ω 2 ); (4) for quaternary matroids M, T M (−1, −1) was found by Vertigan [14], extending a result for graphic/binary matroids by Rosenstiehl and Read [12].…”

Section: Introductionmentioning

confidence: 99%

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On the evaluation at (−ι,ι) of the Tutte polynomial of a binary matroid

Pendavingh

2013

J Algebr Comb

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Vertigan has shown that if M is a binary matroid, then |T M (−ι, ι)|, the modulus of the Tutte polynomial of M as evaluated in (−ι, ι), can be expressed in terms of the bicycle dimension of M. In this paper, we describe how the argument of the complex number T M (−ι, ι) depends on a certain Z/4Z-valued quadratic form that is canonically associated with M. We show how to evaluate T M (−ι, ι) in polynomial time, as well as the canonical tripartition of M and further related invariants.

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“…Vertigan announced that this problem is #P-complete for any fixed field F in [14], but his result remains unpublished.…”

Section: Counting Basesmentioning

confidence: 99%

“…Vertigan [14] shows that if V is a linear subspace of F E and |F| is one of 2, 3, 4, then d(V ) depends only on M(V ).…”

Section: The Bicycle Dimensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the evaluation at (−ι,ι) of the Tutte polynomial of a binary matroid

Pendavingh

2013

J Algebr Comb

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“…We also mention the related paper [5], where the problem of the complexity of the computation of t(M; x, y), for x, y algebraic numbers and M vectorial over a given finite field, is addressed in full generality.…”

mentioning

confidence: 99%

On the evaluation at (j, j 2) of the Tutte polynomial of a ternary matroid

Gioan

Vergnas

2006

J Algebr Comb

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F. Jaeger has shown that up to a ± sign the evaluation at ( j, j 2 ) of the Tutte polynomial of a ternary matroid can be expressed in terms of the dimension of the bicycle space of a representation over G F(3). We give a short algebraic proof of this result, which moreover yields the exact value of ±, a problem left open in Jaeger's paper. It follows that the computation of t( j, j 2 ) is of polynomial complexity for a ternary matroid.In the seminal paper [4] on the complexity of Tutte polynomials, it is shown that the point ( j, j 2 ) and its conjugate ( j 2 , j) are two out of eight 'easy' special points, where 'easy' is intended from a computational point of view. Each of these eight points have remarkable combinatorial interpretations. A result of F. Jaeger [3] relates t( j, j 2 ) and t( j 2 , j) to ternary matroids. Specifically, let E be a finite set, V be a subspace of the vector space G F(3) E , and M(V ) be the matroid on E whose circuits are the inclusion-minimal supports of non-zero vectors of V . Then t(M(V ); j, j 2 ) = ± j |E|+dim V (i √ 3) dim(V ∩V ⊥ ) . Graphs, via graphic matroids, are a special case of ternary matroids. We refer the reader to the introduction of [3] (see also [4]

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Graph Polynomials and Their Applications I: The Tutte Polynomial

Ellis-Monaghan

Merino

2010

Structural Analysis of Complex Networks

117

119

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We give a number of properties and combinatorial interpretations for various evaluations of the Tutte polynomial. We recover colorings, flows, orientations, network reliability, etc., and related polynomials as specializations of the Tutte polynomial. We discuss the coefficients, zeros, and derivatives of the Tutte polynomial, and conclude with a brief discussion of computational complexity. Preliminary NotionsThe graph terminology that we use is standard and generally follows Diestel [Die00]. Graphs may have loops and multiple edges. For a graph G we denote by V (G) its set of vertices and by E(G) its set of edges. An oriented graph, G, also called a digraph, has a direction assigned to each edge. Basic ConceptsWe first recall some of the notions of graph theory most used in this chapter. Two graphsWe denote by κ(G) the number of connected components of a graph G, and by c(G) the number of non-trivial connected components, that is the number of connected components not counting isolated vertices. A graph is k-connected if at least k vertices must be removed to disconnect the graph.A cycle in a graph G is a set of edges e 1 , . . . , e k such that, if e i = (v i , w i ) for 1 ≤ i ≤ k, then w i = v i+1 for 1 ≤ i ≤ k − 1; also w k = v 1 and v i = v j for i = j. A trail is a path that may revisit a vertex, but not retrace an edge. A circuit is a closed trail, and thus a cycle is just a circuit that does not revisit any vertices. In the case of a digraph, the edges of a trail or circuit must be consistently oriented.The dual notion of a cycle is that of cut or cocycle. If {V 1 , V 2 } is a partition of the vertex set, and the set C, consisting of those edges with one end in V 1 and one end in V 2 , is not empty, then C is called a cut. A cycle with one edge is called a loop and a cocycle with one edge is called a cut-edge or bridge. We refer to an edge that is neither a loop nor a bridge as ordinary.A tree is a connected graph without cycles. A forest is a graph whose connected components are all trees. A subgraph H of a graph G is spanning if V (H) = V (G). Spanning trees in connected graphs will play a fundamental role in the theory of the Tutte polynomial. Observe that a loop in a connected graph can be characterized as an edge that is in no spanning tree, while a bridge is an edge that is in every spanning tree.If V ⊆ V (G), then the induced subgraph on V has vertex set V and edge set those edges of G with both endpoints in V . If E ⊆ E(G), then the spanning subgraph induced by E has vertex set V (G) and edge set E .

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Bicycle Dimension and Special Points of the Tutte Polynomial

Cited by 25 publications

References 17 publications

On the evaluation at (−ι,ι) of the Tutte polynomial of a binary matroid

On the evaluation at (−ι,ι) of the Tutte polynomial of a binary matroid

On the evaluation at (j, j 2) of the Tutte polynomial of a ternary matroid

Graph Polynomials and Their Applications I: The Tutte Polynomial

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