The Tutte Polynomial Part I: General Theory

Brylawski, Tom

doi:10.1007/978-3-642-11110-5_3

Cited by 30 publications

(42 citation statements)

References 95 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, m k is the greatest integer m for which the monomial (x − 1) r−k (y − 1) m−k occurs in T (M ; x, y) with non-zero coefficient and f k (m k ) is that coefficient. As shown in Section 5 of [6], f k (s) is derivable from T (M ; x, y) for each s with m k−1 < s ≤ m k .…”

Section: Deriving Matroid Parameters From the G-invariantmentioning

confidence: 97%

The G-invariant and catenary data of a matroid

Bonin

Kung

2018

Advances in Applied Mathematics

View full text Add to dashboard Cite

Section: Deriving Matroid Parameters From the G-invariantmentioning

confidence: 97%

The G-invariant and catenary data of a matroid

Bonin

Kung

2018

Advances in Applied Mathematics

View full text Add to dashboard Cite

“…That t 1,0 = t 0,1 is one of an infinite family of relations among the coefficients of the Tutte polynomial. Brylawski [Bry82] has shown the following:…”

Section: Coefficient Relationsmentioning

confidence: 97%

Graph Polynomials and Their Applications I: The Tutte Polynomial

Ellis-Monaghan

Merino

2010

Structural Analysis of Complex Networks

118

119

View full text Add to dashboard Cite

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 .

show abstract

“…We use the graph theoretic version of Brylawski's tensor product for matroids [Bry11]. We found the following terminology more intuitive in our setting.…”

Section: Graph Inflationsmentioning

confidence: 99%

Exponential Time Complexity of the Permanent and the Tutte Polynomial

Dell

Husfeldt

Marx

et al. 2014

ACM Trans. Algorithms

View full text Add to dashboard Cite

We show conditional lower bounds for well-studied #P-hard problems:• The number of satisfying assignments of a 2-CNF formula with n variables cannot be computed in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph.• The permanent of an n × n matrix with entries 0 and 1 cannot be computed in time exp(o(n)).• The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x, y) in the case of multigraphs, and it cannot be computed in time exp(o(n/ poly log n)) in the case of simple graphs.Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting.

show abstract

The Tutte Polynomial Part I: General Theory

Cited by 30 publications

References 95 publications

The G-invariant and catenary data of a matroid

The G-invariant and catenary data of a matroid

Graph Polynomials and Their Applications I: The Tutte Polynomial

Exponential Time Complexity of the Permanent and the Tutte Polynomial

Contact Info

Product

Resources

About