Dense Digraphs Research Articles

Bounding the Chromatic Number of Dense Digraphs by Arc Neighborhoods

Bounding the Chromatic Number of Dense Digraphs by Arc Neighborhoods

On vertex in-out-antimagic total digraphs

A vertex in-out-antimagic total labeling of a directed graph (digraph) D=(V,A) with n vertices and m arcs is a bijection from the set V∪A to the set of integers {1,2,…,m+n} such that all n vertex in-weights are pairwise distinct and simultaneously all n vertex out-weights are pairwise distinct. The vertex in-weight is the sum of the vertex label and the labels of all incoming arcs and the vertex out-weight is the sum of the vertex label and the labels of all outgoing arcs.In this paper we provide a general way how to label dense digraphs and certain sparse digraphs. Further, we add constructions of the labeling for three large infinite classes of digraphs and we conjecture that all digraphs allow such a labeling.

Interaction graphs of isomorphic automata networks I: Complete digraph and minimum in-degree

An automata network with n components over a finite alphabet Q of size q is a discrete dynamical system described by the successive iterations of a function f:Qn→Qn. In most applications, the main parameter is the interaction graph of f: the digraph with vertex set [n] that contains an arc from j to i if fi depends on input j. What can be said on the set G(f) of the interaction graphs of the automata networks isomorphic to f? It seems that this simple question has never been studied. Here, we report some basic facts. First, we prove that if n≥5 or q≥3 and f is neither the identity nor constant, then G(f) always contains the complete digraph Kn, with n2 arcs. Then, we prove that G(f) always contains a digraph whose minimum in-degree is bounded as a function of q. Hence, if n is large with respect to q, then G(f) cannot only contain Kn. However, we prove that G(f) can contain only dense digraphs, with at least ⌊n2/4⌋ arcs.

Coloring Dense Digraphs

The chromatic number of a digraph D is the minimum number of acyclic subgraphs covering the vertex set of D. A tournament H is a hero if every H-free tournament T has chromatic number bounded by a function of H. Inspired by the celebrated Erdős-Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph H is a superhero if every H-free digraph D has chromatic number bounded by a function of H and α(D), the independence number of the underlying graph of D. We prove here that a digraph is a superhero if and only if it is a hero, and hence characterize all superheroes. This answers a question of Aboulker, Charbit and Naserasr.

From Subkautz Digraphs to Cyclic Kautz Digraphs

The Kautz digraphs K(d, ℓ) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related to these, the cyclic Kautz digraphs CK(d, ℓ) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to the cyclic Kautz digraphs by introducing the family of the subKautz digraphs sK(d, ℓ), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d, ℓ) and CK(d, ℓ). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d, ℓ) and CK(d, ℓ) are maximally vertex-connected and super-edge-connected. Whereas K(d, ℓ) are optimal with respect to the diameter, we show that sK(d, ℓ) and CK(d, ℓ) are optimal with respect to the mean distance, whose exact values are given for both families when ℓ = 3. Finally, we provide a lower bound on the girth of CK(d, ℓ) and sK(d, ℓ).

Coloring dense digraphs

The chromatic number of a digraph D is the minimum number of acyclic subgraphs covering the vertex set of D. A tournament H is a hero if every H-free tournament T has chromatic number bounded by a function of H. Inspired by the celebrated Erdős–Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. Motivated by a question of the first author and Colin McDiarmid, we study digraphs which we call superheroes. A digraph H is a superhero if every H-free digraph D has chromatic number bounded by a function of H and α(D), the independence number of the underlying graph of D. We prove here that a digraph is a superhero if and only if it is a hero, and hence characterize all superheroes. This answers a question of Pierre Aboulker.

The spectra of subKautz and cyclic Kautz digraphs

Kautz digraphs K(d,ℓ) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz CK(d,ℓ) and the subKautz sK(d,2) digraphs were recently introduced by Böhmová, Huemer and the author.In this paper we propose a new method to obtain the complete spectra of subKautz sK(d,2) and cyclic Kautz CK(d,3) digraphs, for all d≥3, through the Hoffman–McAndrew polynomial and regular partitions. This approach can be useful to find the spectra of other families of digraphs with high regularity.

Iterated line digraphs are asymptotically dense

We show that the line digraph technique, when iterated, provides dense digraphs, that is, with asymptotically large order for a given diameter (or with small diameter for a given order). This is a well-known result for regular digraphs. In this note we prove that this is also true for non-regular digraphs.

An improved fully polynomial randomized approximation scheme (FPRAS) for counting the number of Hamiltonian cycles in dense digraphs

We propose an improved algorithm for counting the number of Hamiltonian cycles in a directed graph. The basic idea of the method is sequential acceptance/rejection, which is successfully used in approximating the number of perfect matchings in dense bipartite graphs. As a consequence, a new ratio of the number of Hamiltonian cycles to the number of 1-factors is proposed. Based on this ratio, we prove that our algorithm runs in expected time of O ( n 8.5 ) for dense problems. This improves the Markov chain Monte Carlo method, the most powerful existing method, by a factor of at least n 4.5 ( log n ) 4 in running time. This class of dense problems is shown to be nontrivial in counting, in the sense that they are #P-Complete.

A line digraph of a complete bipartite digraph

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore-like bound in terms of its diameter k and the maximum outdegrees ( d 1 , d 2 ) of its partite sets of vertices. In this work, we define a family of dense digraphs, the diameter of which is not more than 1, comparable with that of the Moore bipartite digraph of the same order and maximum degree. Furthermore, some of its properties are given, such as the connectivity, spectrum and so on.

Cycles in dense digraphs

Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let β(G) denote the size of the smallest subset X ⊆ E(G) such that G∖X has no directed cycles, and let γ(G) be the number of unordered pairs {u, v} of vertices such that u, v are nonadjacent in G. It is easy to see that if γ(G) = 0 then β(G) = 0; what can we say about β(G) if γ(G) is bounded?

A dynamic topological sort algorithm for directed acyclic graphs

We consider the problem of maintaining the topological order of a directed acyclic graph (DAG) in the presence of edge insertions and deletions. We present a new algorithm and, although this has inferior time complexity compared with the best previously known result, we find that its simplicity leads to better performance in practice. In addition, we provide an empirical comparison against the three main alternatives over a large number of random DAGs. The results show our algorithm is the best for sparse digraphs and only a constant factor slower than the best on dense digraphs.

Polynomially Improved Efficiency for Fast Parallel Single-Source Lexicographic Depth-First Search, Breadth-First Search, and Topological-First Search

Although lexicographic (lex) variants of greedy algorithms are often P -complete, NC -algorithms are known for the following lex-search problems: lexicographic depth-first search (lex-dfs) for dags [12], [17], lexicographic breadth-first search (lex-bfs) for digraphs [12], [17], and lexicographic topological-first search (lex-tfs) for dags [12]. For the all-sources version of the problem for dense digraphs, the lex-dfs (lex-bfs, lex-tfs) in [12] is (within a log factor of) work-optimal with respect to the all-sources sequential solution that performs a dfs (bfs, tfs) from every vertex. By contrast, to solve the single-source lexicographic version on inputs of size n , all known NC -algorithms perform work that is at least an n factor away from the work performed by their sequential counterparts.

Super edge-connectivity of dense digraphs and graphs

Super-λ is a more refined network reliability index than edge-connectivity. A graph is super-λ if every minimum edge-cut set is trivial (the set of edges incident at a node with the minimum degree δ). This paper establishes the relation between diameter and super-λ: enlarging the order n under the given maximum degree Δ and diameter D not only maximizes edge-connectivity, but also minimizes the number of minimum edge-cut sets, thus attaining super-λ. The following sufficient conditions for a diagraph and graph G to be super-λ are derived. • • Digraph G is super-λ if n>δ(( Δ D−1−1) (Δ−1) +1)+Δ D−1 . • • Graph G is super-λ if n>δ((( Δ−1) D−1−1) (Δ−2) +1)+(Δ−1) D−1 . These conditions are the best possible. From these, the de Bruijn digraph B( d,D),the Kautz digraph K( d, D), and most of the densest known graphs (listed in [3, 10]) are shown to be super-λ Also, the digraph G B ∗( n,d) proposed in [25] as a maximally connected d-regular digraph with quasiminimal diameter (at most one larger than the lower bound) is proved to be super-λ for any d>2 and any order n> d 3.

Parametric shortest path algorithms with an application to cyclic staffing

Let G=( V, E) be a digraph with n vertices including a special vertex s. Let E′ ⊆ E be a designated subset of edges. For each e ϵ E there is an associated real number ƒ 1(e) . Furthermore, let ƒ 2(e)=1 if e ϵ E′ and ƒ 2(e)=0 if e ϵ E − E′. The length of edge e is ƒ 1(e)− λƒ 2(e) , where λ is a parameter that takes on real values. Thus the length varies additively in λ for each edge of E′. We shall present two algorithms for computing the shortest path from s to each vertex υ ϵ V parametrically in the parameter λ, with respective running times O( n 3) and O( n| E|log n). For dense digraphs the running time of the former algorithm is comparable to the fastest (non-parametric) shortest path algorithm known. This work generalizes the results of Karp [2] concerning the minimum cycle mean of a digraph, which reduces to the case that E′= E. Furthermore, the second parametric algorithm may be used in conjunction with a transformation given by Bartholdi, Orlin, and Ratliff [1] to give an O( n 2 log n) algorithm for the cyclic staffing problem.