This paper presents a state-of-the-art algorithm for the vertex enumeration problem of arrangements, which is based on the proposed new pivot rule, called the Zero rule. The Zero rule possesses several desirable properties: i) It gets rid of the objective function; ii) Its terminal satisfies uniqueness; iii) We establish the if-and-only if condition between the Zero rule and its valid reverse, which is not enjoyed by earlier rules; iv) Applying the Zero rule recursively definitely terminates in $d$ steps, where $d$ is the dimension of input variables. Because of so, given an arbitrary arrangement with $v$ vertices of $n$ hyperplanes in $\mathbb{R}^d$, the algorithm's complexity is at most $\mathcal{O}(n^2d^2v)$ and can be as low as $\mathcal{O}(nd^4v)$ if it is a simple arrangement, while Moss' algorithm takes $\mathcal{O}(nd^2v^2)$, and Avis and Fukuda's algorithm goes into a loop or skips vertices because the if-and-only-if condition between the rule they chose and its valid reverse is not fulfilled. Systematic and comprehensive experiments confirm that the Zero rule not only does not fail but also is the most efficient.
We give a complete formula for the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $ 1\leq i, j, k, l\leq n$. The formula is obtained by associating hyperplane arrangements with graphs, and then enumerating central graphs via generating functions for the number of bipartite graphs of given order, size and number of connected components.
The number of flats of a hyperplane arrangement is considered as a generalization of the Bell number and the Stirling number of the second kind. Robert Gill gave the exponential generating function of the number of flats of the extended Catalan arrangements, using species. In this article, we introduce the species of flats of the extended Catalan and Shi arrangements and they are given by iterated substitution of species of sets and lists. Moreover, we enumerate the flats of these arrangements in terms of infinite matrices.
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated discriminantal arrangement on the other hand. The type of hyperplane arrangements considered and the isomorphism classes have been defined precisely. As a consequence we enumerate such isomorphism classes by computing the characteristic polynomial of the discriminantal arrangement. With a certain restriction, the enumerated value is shown to be independent of the discri…
Universal cycle for $k$-permutations is a cyclic arrangement in which each $k$-permutation appears exactly once as $k$ consecutive elements. Enumeration problem of universal cycles for $k$-permutations is discussed and one new enumerating method is proposed in this paper. Accurate enumerating formulae are provided when $k=2,3$.
A probabilistic bijection between twenty-vertex configurations with a free west boundary and Gelfand-Tsetlin patterns avoiding three equal entries in a row
Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson. Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the maximum number of minimal connected vertex covers of a graph is at most 1.8668^n, and these can be enumerated in time O(1.8668^n). For graphs of chordality at most 5, we are able to give a bette…
Over the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorithms for high-dimensional polyhedra given by inequalities. It is a famous and long standing open question in polyhedral theory and computational geometry whether the vertices of a polytope (bounded polyhedron), described by a set of linear constraints, can be enumerated in total polynomial time. In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra $P = \{x : A…
Let $ G $ be a simple graph of $ \ell $ vertices $ \{1, \dots, \ell \} $ with edge set $ E_{G} $. The graphical arrangement $ \mathcal{A}_{G} $ consists of hyperplanes $ \{x_{i}-x_{j}=0\} $, where $ \{i, j \} \in E_{G} $. It is well known that three properties, chordality of $ G $, supersolvability of $ \mathcal{A}_{G} $, and freeness of $ \mathcal{A}_{G} $ are equivalent. Recently, Richard P. Stanley introduced $ ψ$-graphical arrangement $ \mathcal{A}_{G, ψ} $ as a generalization of graphical arrangements. Lili Mu and Stanley characterized the supersolvability of the $ ψ$-graphical arrangemen…
Let $D$ be a connected weighted digraph. The relation between the vertex weighted complexity (with a fixed root) of the line digraph of $D$ and the edge weighted complexity (with a fixed root) of $D$ has been given in (L. Levine, Sandpile groups and spanning trees of directed line graphs, J. Combin. Theory Ser. A 118 (2011) 350-364) and, independently, in (S. Sato, New proofs for Levine's theorems, Linear Algebra Appl. 435 (2011) 943-952). In this paper, we obtain a relation between the vertex weighted complexity of the middle digraph of $D$ and the edge weighted complexity of $D$. Particularl…
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $γ(G)$, is the domination number of $G$. A dominating set $D$ is an accurate dominating set of $G$, if no $|D|$-element subset of $V\setminus D$ is a dominating set of $G$. The accurate domination number, $γ_a(G)$, is the cardinality of a smallest accurate dominating set $D$. In this paper, after presenting preliminaries, we count the number of accurate dominating sets of …
We address the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses sit next to each other and no k consecutive persons are of the same gender. While the case of k=2 corresponds to the classical problème des ménages with a well-studied solution, no closed-form expression for the number of seating arrangements is known when k>=3. We propose a novel approach for this type of problems based on enumeration of walks in certain algebraically weighted de Bruijn graphs. Our approach leads to new expressions for the menage numbers and their expo…
Bernardi has given a general formula for the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that each set of boxed trees which share an underlying (rooted labeled plane) tree contributes 0, +1, or -1 to this sum, and we give an algorithm for computing this value. For Ish-type arrangements, we further construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of (rooted labeled plane) trees. We conclude by explicitly enumerating the trees corresponding to the regions of Ish-type arrangements which…
In this paper, we provide an algorithm for traversing geometric graphs which visits all vertices, and reports every vertex and edge exactly once. To achieve this, we combine a given geometric graph $G$ with the integer lattice, seen as a graph, in such a way that the resulting hypothetical graph can be traversed using the algorithm in \cite{Chavez}. To overcome the problem with hypothetical vertices and edges, we develop an algorithm for visiting any $k$th neighborhood of a vertex in a graph straight-line drawn in the plane using $O(\log k)$ memory. The memory needed to complete the traversal …
We prove that for any fixed $k$, the probability that a random vertex of a random increasing plane tree is of rank $k$, that is, the probability that a random vertex is at distance $k$ from the leaves, converges to a constant $c_k$ as the size $n$ of the tree goes to infinity. {\color{blue} We prove that $1-\sum_{j\le k} c_k<\tfrac{3^{k+1}}{(2k+1)!}$, so that the tail of the limiting rank distribution is super-exponentially narrow. We prove that the latter property holds uniformly for all finite $n$ as well.} More generally, we prove that the ranks of a finite uniformly random set of vertices …
We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex- and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular *-) semigroups, and give several tables of calculated values.
We introduce a new family of hyperplane arrangements inspired by the homogenized Linial arrangement (which was recently introduced by Hetyei), and show that the intersection lattices of these arrangements are isomorphic to the bond lattices of Ferrers graphs. Using recent work of Lazar and Wachs we are able to give combinatorial interpretations of the characteristic polynomials of these arrangements in terms of permutation enumeration. For certain infinite families of these hyperplane arrangements, we are able to give generating function formulas for their characteristic polynomials. To do so,…
An { \em arrangement} of an ordered pair $(G_A, G_M)$ of graphs is defined as a function $f$ from $V(G_A)$ to $V(G_M)$ such that, for each vertex $c$ of $G_M$ , the vertex-set $f^{-1}(c)$ of $G_A$ either is $\emptyset$ (the case when $c \not\in f(V(G_A))$ ) or induces a connected subgraph of $G_A$ and that the family $\{f^{-1}(y) : y \in V(G_M), f^{-1}(y) \neq \emptyset\}$ is a partition of $V(G_A)$ . Let $f$ be an arrangement of $(G_A, G_M)$ , let $pq$ be an edge of $G_M$ and let $U$ be a subset of $f^{-1}(p)$ such that each of the three graphs $G_A[U]$ , $G_A[f^{-1}(p)\setminus U]$ and $G_A[…
Estimating the frequency of sub-graphs is of importance for many tasks, including sub-graph isomorphism, kernel-based anomaly detection, and network structure analysis. While multiple algorithms were proposed for full enumeration or sampling-based estimates, these methods fail in very large graphs. Recent advances in parallelization allow for estimates of total sub-graphs counts in very large graphs. The task of counting the frequency of each sub-graph associated with each vertex also received excellent solutions for undirected graphs. However, there is currently no good solution for very larg…
A Generalized Enumeration of Labeled Trees and Reverse Prüfer Algorithm
$[n]$上的树$T$的{ \em leader}是在$T$中没有更小后代的顶点。Gessel和Seo证明了$$\sum_{T \in \mathcal{T}_n}u^\text{(# of leaders in $T$)} c^\text{(degree of 1 in $T$)}=u P_{n-1}(1,u,cu),$$，它是Cayley公式的推广，其中$\mathcal{T}_n$是$[n]$和$$P_n(a,b,c)=c\prod_{i=1}^{n-1}(ia+(n-i)b+c).$$上的树的集合。利用Prüfer码的一种变体，称为{ \em RP-码}，我们给出了Gessel和Seo公式的一个简单的双射证明。
Mirror graphs were introduced by Brešar et al. in 2004 as an intriguing class of graphs: vertex-transitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article we settle the structure of mirror graphs by characterizing them as precisely the Cayley graphs of the finite Coxeter groups or equivalently the tope graphs of reflection arrangements - well understood and classified structures. We provide a polynomial algorithm for their recognition.
A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main resu…
We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In part…
