Acyclic Orientations Research Articles

A note on a counting problem arising in percolation theory

We investigate an esoteric enumeration problem on graphs which arises from a physical model related to the continuum percolation theory. It counts signed objects, but surprisingly, we are able to find an equivalent count of positive signs only, namely, restricted acyclic orientations. This reduces the counting effort, but shows on the other hand that it is a #P-complete problem.

A generalized chromatic polynomial, acyclic orientations with prescribed sources and sinks, and network reliability

Suppose G=( V, E) is a graph and K, K′, K″ are subsets of V such that K⊆ K′ ∩ K″. We introduce and study a polynomial P( G, K, K′, K″; λ) in λ. This polynomial coincides with the classical chromatic polynomial P( G; λ) when K= V. The results of this paper generalize Whitney's characterizations of the coefficients of P( G; λ) and the work of Stanley on acyclic orientations. Furthermore, we establish a connection between a family of polynomials associated with network reliability and a family of polynomials associated with P( G, K, K′, K″; λ).

Ramsey properties of orientations of graphs

AbstractA classical result of Rödl (in Graphs, Hypergraphs and Block Systems, 1976, pp. 211 to 220) says that for any acyclic digraph D there is a graph G = G(D) such that every acyclic orientation of G contains an induced copy of D. A recent result of Rödl and Winkler (SIAM J. Discrete Math., 2: 402–406, 1989) implies that there is such a graph G of order O(n3 (log n)2), where n = |D| is the order of D. Here we show by probabilistic means that almost every graph Gn of order [(4/9)n2n/2] has the property that every acyclic orientation of Gn contains an induced copy of every acyclic digraph on n vertices. We also show that the order of any such graph Gn has to be greater than (1/10)n2n/2. Thus our results are essentially best possible. Moreover, we show that there are Paley graphs of order O(n24n) having the property above. We also consider some problems raised by Cochand and Duchet (Irregularities of Partitions, Springer‐Verlag, Berlin, 1989, pp. 39–46.) on related topics. © 1993 John Wiley & Sons, Inc.

Perfectly orderable graphs and almost all perfect graphs are kernelM-solvable

A graph is perfectly orderable if and only if it admits an acyclic orientation which does not contain an induced subgraph with verticesa, b, c, d and arcsab, bc, dc. Further a graph is called kernelM-solvable if for every direction of the edges (here pairs of symmetric, i.e. reversible, arcs are allowed) such that every directed triangle possesses at least two pairs of symmetric arcs, there exists a kernel, i.e. an independent setK of vertices such that every other vertex sends some arc towardsK. We prove that perfectly orderable graphs are kernelM-solvable. Using a deep result of Promel and Steger we derive that almost all perfect graphs are kernelM-solvable.

Planar orientations with low out-degree and compaction of adjacency matrices

We consider the problem of orienting the edges of a planar graph in such a way that the out-degree of each vertex is minimized. If, for each vertex v, the out-degree is at most d, then we say that such an orientation is d-bounded. We prove the following results: • Each planar graph has a 5-bounded acyclic orientation, which can be constructed in linear time. • Each planar graph has a 3-bounded orientation, which can be constructed in linear time. • A 6-bounded acyclic orientation, and a 3-bounded orientation, of each planar graph can each be constructed in parallel time O(log n log ∗ n) on an EREW PRAM, using O( n/log n log ∗ n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time.

Some Alternate Characterizations of Reliability Domination

An important problem in reliability theory is to determine the reliability of a system from the reliability of its components. If E is a finite set of components, then certain subsets of E are prescribed to be the operating states of the system. A formation is any collection F of minimal operating states whose union is E. Reliability domination is defined as the total number of odd cardinality formations minus the total number of even cardinality formations. The purpose of this paper is to establish some new results concerning reliability domination. In the special case where the system can be identified with a graph or digraph, these new results lead to some new graph-theoretic properties and to simple proofs of certain known theorems. The pertinent graph-theoretic properties include spanning trees, acyclic orientations, Whitney's broken cycles, and Tutte's internal activity associated with the chromatic polynomial.

Concurrency in heavily loaded neighborhood-constrained systems

Let G be a connected undirected graph in which each node corresponds to a process and two nodes are connected by an edge if the corresponding processes share a resource. We consider distributed computations in which processes are constantly demanding all of their resources in order to operate, and in which neighboring processes may not operate concurrently. We advocate that such a system is general enough for representing a large class of resource-sharing systems under heavy load. We employ a distributed scheduling mechanism based on acyclic orientations of G and investigate the amount of concurrency that it provides. We show that this concurrency is given by a number akin to G 's chromatic and multichromatic numbers, and that, among scheduling schemes which require neighbors in G to alternate in their turns to operate, ours is the one that potentially provides the greatest concurrency. However, we also show that the decision problem corresponding to optimizing concurrency is NP -complete.

Neural Networks, Acyclic Orientations of the Hypercube, and Sets of Orthogonal Vectors

Recent models in the theory of neural networks suggest the possibility of constructing new combinatorial invariants by associating to families of subsets of an n-set orientations of the hypercube via “energy functions.” Here, we restrict ourselves to the Hopfield model and its higher-order versions with families of orthogonal binary vectors and homogeneous polynomial functions, corresponding to sums of outerproducts of degree d, and investigate several properties of the corresponding orientations. In particular, the stability of the vectors in the family and of those in the orthogonal space is analyzed. The existence of significant differences of behavior according to the congruences modulo 4 of d is shown.

Graph colorings and acyclic orientations

This paper presents new combinatorial proofs of two identities due to R. Stanley relating the chromatic polynomial and acyclic orientations of a graph. In addition, using elementary means, explicit formulae for the generating functions of the chromatic numbers and the number of color compatible acyclic orientations are derived. These formulae immediately show a reciprocity law concerning the generating functions.

The chromatic generating function

Let G=(V E) be a graph of order n and Pg (x) its chromatic polynomial. It has been shown elsewhere that the chromatic generating function is rational and of the form In this note, we give an explicit formula for the coefficients of p(t) and show that these coefficients count the acyclic orientations of various reduced graphs obtained from the graph G. This establishes combinatorially that these coefficients are positive integers.

Graphical Eulerian numbers and chromatic generating functions

In this paper some further properties of the coefficients of a chromatic generating function introduced by Linial are proved. A combinatorial interpretation of these numbers is given by specializing some results of Stanley on posets to surjective n-colorings of a graph G of order n compatible with linear orders on V( G) which extend acyclic orientations of G. If G is composed from n isolated vertices these coefficients are classical Eulerian numbers A( n, k).

Hard Enumeration Problems in Geometry and Combinatorics

A number of natural enumeration problems in geometry and combinatorics are shown to be complete in the class # P introduced by Valiant. Among others this is established for the numeration of vertices and of facets of a polytope, acyclic orientations of a graph and satisfying assignments of implicative boolean formulas.

Bijective proofs of two broken circuit theorems

AbstractWe prove, by means of explicit bijections, theorems of Whitney and Stanley that express the coefficients of the chromatic polynomial of a graph G and the number of acyclic orientations of G in terms of numbers of sets of edges that contain no broken circuits of G.

The chromatic polynomial of an unlabeled graph

We investigate the chromatic polynomial χ( G, λ) of an unlabeled graph G. It is shown that χ(G, λ) = ( 1 |A(g)| ) Σ π ∈ A(g) χ(g, π, λ) , where g is any labeled version of G, A( g) is the automorphism group of g and χ( g, π, λ) is the chromatic polynomial for colorings of g fixed by π. The above expression shows that χ( G, λ) is a rational polynomial of degree n = | V( G)| with leading coefficient 1 |A(g)| . Though χ( G, λ) does not satisfy chromatic reduction, each polynomial χ( g, π, λ) does, thus yielding a simple method for computing χ( G, λ). We also show that the number N( G) of acyclic orientations of G is related to the argument λ = −1 by the formula N(G) = ( 1 |A(g)| ) Σ π ∈ A(g)(−1) s(π) χ(g, π, −1) , where s( π) is the number of cycles of π. This information is used to derive Robinson's (“Combinatorial Mathematics V” (Proc. 5th Austral. Conf. 1976), Lecture Notes in Math. Vol. 622, pp. 28–43, Springer-Verlag, New York/Berlin, 1977) cycle index sum equations for counting unlabeled acyclic digraphs.

Network reliability and acyclic orientations

AbstractLet G = (V, E) be an undirected graph with V = n perfectly reliable vertices and E = m edges that work with probability pe and fail with probability qe for all e /in E. The network reliability problem considered here is to calculate RK[G] = Prob [there exists a path of working edges between every pair of a set K of k vertices]. Backtrack algorithms for this problem are presented. The complexity of these algorithms is discussed when applied to the all‐terminal network reliability problem where K = V. Strategies are presented which minimize the size of the search structures generated. The main theorems state that when these strategies are used, the size of the search structures may be measured by counting spanning trees or acyclic orientations.

The acyclic sets of an oriented matroid

The simplicial complex of acyclic sets of an oriented matroid is studied. The complex is shown to have the homotopy type of a sphere. As an application, using Euler characteristic arguments, the number of acyclic orientations is computed.

Transversals of circuits and acyclic orientations in graphs and matroids

Soit M un matroide oriente sans boucles et soit A un ensemble d'elements de M. Alors les proprietes suivantes sont equivalentes: 1) A rencontre chaque circuit positif de M et est minimal pour l'inclusion avec cette propriete; 2) A M est acyclique et A est minimal pour l'inclusion avec cette propriete

A partial order on the regions of 𝑅ⁿ dissected by hyperplanes

We study a partial order on the regions of R n {{\mathbf {R}}^n} dissected by hyperplanes. This includes a computation of the Möbius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs.

Parallel sorting

We show that if n is sufficiently large then there is a graph G of order n with ⌊n 3 2 log n⌋ edges such that the transitive closure of every acyclic orientation of G has at least n 2 −n 3 2 log n edges. A consequence of this is that with ⌊n 3 2 log n⌋ parallel processors n objects may be sorted in two time intervals. This improves considerably some results of Häggkvist and Hell. We prove similar assertions about sorting with only d -step implications.

On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs

The doubly indexed Whitney numbers of a finite, ranked partially ordered set L L are (the first kind) w i j = ∑ { μ ( x i , x j ) : x i , x j ∈ L {w_{ij}} = \sum {\{ \mu ({x^i},{x^j}):{x^i},{x^j} \in L} with ranks i , j } i,j\} and (the second kind) W i j = {W_{ij}} = the number of ( x i , x j ) ({x^i},{x^j}) with x i ⩽ x j {x^i} \leqslant {x^j} . When L L has a 0 0 element, the ordinary (simply indexed) Whitney numbers are w j = w 0 j {w_j} = {w_{0j}} and W j = W 0 j = W j j {W_j} = {W_{0j}} = {W_{jj}} . Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of k k -dimensional faces for any k k , of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope P P inside the visible boundary as seen from a distant point on a generating line of P P . The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley’s theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly q q sources (generalizing Rényi’s enumeration of permutations with q q "outstanding" elements). The number of totally cyclic orientations of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph’s having a unique source.