Graham R. Brightwell

Counting linear extensions

We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #P-complete, settling a long-standing open question. This result is contrasted with recent work giving randomized polynomial-time algorithms for estimating the number of linear extensions.One consequence of our main result is that computing the volume of a rational polyhedron is strongly #P-hard. We also show that the closely related problems of determining the average height of an element x of a give poset, and of determining the probability that x lies below y in a random linear extension, are #P-complete.

Counting linear extensions is #P-complete

We show that the problem of counting the number of linear extensions of a given partially ordered set is #P-complete. This settles a long-standing open question and contrssts with recent results giving randomized polynomial-time algorit hms for estimating the number of linear extensions. One consequence is that computing the volume of a rational polyhedron is strongly #P-hard. We also show that the closely related problems of determining the average height of an element c of a given poset, and of determining the probability that z lies below y in a random linear extension, are #P-complete.

Representations of planar graphs

This paper shows that every 3-connected planar graph G can be represented as a collection of circles, one circle representing each vertex and each face, so that, for each edge of G, the four circles representing the two endpoints and the two neighboring faces meet at a point, and furthermore the vertex-circles cross the face-circles at right angles. This extends a result of W. Thurston [The Geometry and Topology of Three Manifolds, unpublished] and, independently, Andreev. From this we deduce two corollaries: (1) The partial order formed by taking the vertices, edges, and bounded faces of G, ordered by inclusion, is a circle order; (2) One can represent G and its dual simultaneously in the plane with straight-line edges so that the edges of G cross the dual edges at right angles. This answers a question first asked by W. Tutte [Proc. LMS, 13 (3) (1963), pp. 743–768].

Graph Homomorphisms and Phase Transitions

We model physical systems with “hard constraints” by the space Hom(G, H) of homomorphisms from a locally finite graph G to a fixed finite constraint graph H. For any assignment ? of positive real activities to the nodes of H, there is at least one Gibbs measure on Hom(G, H); when G is infinite, there may be more than one. When G is a regular tree, the simple, invariant Gibbs measures on Hom(G, H) correspond to node-weighted branching random walks on H. We show that such walks exist for every H and ?, and characterize those H which, by admitting more than one such construction, exhibit phase transition behavior.

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let TG(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n‐vertex graph G and vertices x and y for which TG(x, y) is maximized. the extremal graph consists of a clique on ⌊(2n + 1)/3⌋) (or ⌈)(2n − 2)/3⌉) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time TG(x, y) to reach y from x in this graph is approximately 4n3/27.

On specifying Boolean functions by labelled examples

Abstract We say a function t in a set H of 0, 1-valued functions defined on a set X is specified by S⊆ X if the only function in H which agrees with t on S is t itself. The specification number of t is the least cardinality of such an S. For a general finite class of functions, we show that the specification number of any function in the class is at least equal to a parameter from Romanik and Smith (1990) known as the testing dimension of the class. We investigate in some detail the specification numbers of functions in the set of linearly separable Boolean functions of n variables — those functions f such that f−1(0) and f−1(1) can be separated by a hyperplane. We present general methods for finding upper bounds on these specification numbers and we characterise those functions which have largest specification number. We obtain a general lower bound on the specification number and we show that for all nested functions, this lower bound is attained. We give a simple proof of the fact that for any linearly separable Boolean function, there is exactly one set of examples of minimal cardinality which specifies the function. We discuss those functions which have limited dependence, in the sense that some of the variables are redundant (that is, there are irrelevant attributes), giving tight upper and lower bounds on the specification numbers of such functions. We then bound the average, or expected, number of examples needed to specify a linearly separable Boolean function. In the final section of the paper, we address the complexity of computing specification numbers and related parameters.

Gibbs Measures and Dismantlable Graphs

We model physical systems with “hard constraints” by the space Hom(G, H) of homomorphisms from a locally finite graph G to a fixed finite constraint graph H. Two homomorphisms are deemed to be adjacent if they differ on a single site of G. We investigate what appears to be a fundamental dichotomy of constraint graphs, by giving various characterizations of a class of graphs that we call dismantlable. For instance, H is dismantlable if and only if, for every G, any two homomorphisms from G to H which differ at only finitely many sites are joined by a path in Hom(G, H). If H is dismantlable, then, for any G of bounded degree, there is some assignment of activities to the nodes of H for which there is a unique Gibbs measure on Hom(G, H). On the other hand, if H is not dismantlable (and not too trivial), then there is some r such that, whatever the assignment of activities on H, there are uncountably many Gibbs measures on Hom(Tr, H), where Tr is the (r+1)-regular tree.

Cycles through specified vertices

AbstractRecently, various authors have obtained results about the existence of long cycles in graphs with a given minimum degreed. We extend these results to the case where only some of the vertices are known to have degree at leastd, and we want to find a cycle through as many of these vertices as possible. IfG is a graph onn vertices andW is a set ofw vertices of degree at leastd, we prove that there is a cycle through at least

On exact specification by examples

\left\lceil {\frac{w}{{\left\lceil {{n \mathord{\left/ {\vphantom {n d}} \right. \kern-\nulldelimiterspace} d}} \right\rceil - 1}}} \right\rceil