Iterated Line Digraphs Research Articles

Distance‐layer structure of the De Bruijn and Kautz digraphs: Analysis and application to deflection routing

AbstractIn this article, we present a detailed study of the reach distance‐layer structure of the De Bruijn and Kautz digraphs, and we apply our analysis to the performance evaluation of deflection routing in De Bruijn and Kautz networks. Concerning the distance‐layer structure, we provide explicit polynomial expressions, in terms of the degree of the digraph, for the cardinalities of some relevant sets of this structure. Regarding the application to defection routing, and as a consequence of our polynomial description of the distance‐layer structure, we formulate explicit expressions, in terms of the degree of the digraph, for some probabilities of interest in the analysis of this type of routing. De Bruijn and Kautz digraphs are fundamental examples of digraphs on alphabet and iterated line digraphs. If the topology of the network under consideration corresponds to a digraph of this type, we can perform, in principle, a similar vertex layer description.

Hamiltonian index of directed multigraph

The index of a property P for a directed multigraph D is the smallest nonnegative integer k such that the iterated line digraph Lk(D) has the property P. Let e(D) denote the eulerian index of D and h(D) denote the hamiltonian index of D. Directed multigraphs families F and H are defined such that a directed multigraph D has a finite value e(D) if and only if D∈F, and D has a finite value h(D) if and only if D∈F∪H. Furthermore, the values of the hamiltonian indices for members in F∪H are determined. In addition, line digraph stable properties are investigated, and sufficient and necessary conditions are obtained for a subfamily of strong directed multigraphs in which being eulerian and being hamiltonian are line digraph stable.

On d-Fibonacci digraphs

The d -Fibonacci digraphs F ( d , k ) , introduced here, have the number of vertices following some generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their nice properties. For instance, F ( d , k ) has diameter d + k − 2 and is semi-pancyclic; that is, it has a cycle of every length between 1 and ℓ , with ℓ ∈ {2 k − 2, 2 k − 1} . Moreover, it turns out that several other numbers of F ( d , k ) (of closed l -walks, classes of vertices, etc.) also follow the same linear recurrences as the numbers of vertices of the d -Fibonacci digraphs.

Identifying codes in line digraphs

Given an integer $\ell\ge 1$, a $(1,\le \ell)$-identifying code in a digraph is a dominating subset $C$ of vertices such that all distinct subsets of vertices of cardinality at most $\ell$ have distinct closed in-neighbourhood within $C$. In this paper, we prove that every $k$-iterated line digraph of minimum in-degree at least 2 and $k\geq2$, or minimum in-degree at least 3 and $k\geq1$, admits a $(1,\le \ell)$-identifying code with $\ell\leq2$, and in any case it does not admit a $(1,\le \ell)$-identifying code for $\ell\geq3$. Moreover, we find that the identifying number of a line digraph is lower bounded by the size of the original digraph minus its order. Furthermore, this lower bound is attained for oriented graphs of minimum in-degree at least 2.

Zero forcing in iterated line digraphs

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks.In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications.

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.

Sequence mixed graphs

A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and iterated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed digraph by identifying some vertices of a certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.

Layer structure of De Bruijn and Kautz digraphs. An application to deflection routing

In the main part of this paper we present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the well-known De Bruijn and Kautz digraphs. More precisely, given a vertex v, let Si⋆ (v) be the set of vertices at distance i from v. We show that |Si⋆(v)|=di−ai−1di−1−⋯−a1d−a0, where d is the degree of the digraph and the coefficients ak∈{0,1} are explicitly calculated. Analogously, let w be a vertex adjacent from v such that Si⋆(v)∩Sj⁎(w)≠∅ for some j. We prove that |Si⋆(v)∩Sj⁎(w)|=di−bi−1di−1−…−b1d−b0, where the coefficients bt∈{0,1} are determined from the coefficients ak of the polynomial expression of |Si⋆(v)|. An application to deflection routing in De Bruijn and Kautz networks serves as motivation for our study. It is worth-mentioning that our analysis can be extended to other families of digraphs on alphabet or to general iterated line digraphs.

A Note on the Order of Iterated Line Digraphs

AbstractGiven a digraph G, we propose a new method to find the recurrence equation for the number of vertices of the k‐iterated line digraph , for , where . We obtain this result by using the minimal polynomial of a quotient digraph of G.

The Diameter of Cyclic Kautz Digraphs

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d,ℓ) and it is derived from the Kautz digraphs K(d,ℓ).It is well-known that the Kautz digraphs K(d,ℓ) have the smallest diameter among all digraphs with their number of vertices and degree. Here we define the cyclic Kautz digraphs CK(d,ℓ), whose vertices are labeled by all possible sequences a1…aℓ of length ℓ, such that each character ai is chosen from an alphabet containing d+1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that a1≠aℓ. The cyclic Kautz digraphs CK(d,ℓ) have arcs between vertices a1a2…aℓ and a2…aℓaℓ+1, with a1≠aℓ, a2≠aℓ+1, and ai≠ai+1 for i=1,…,ℓ−1. The cyclic Kautz digraphs CK(d,ℓ) are subdigraphs of the Kautz digraphs K(d,ℓ).We give the main parameters of CK(d,ℓ) (number of vertices, number of arcs, and diameter). Moreover, we construct the modified cyclic Kautz digraphs MCK(d,ℓ) to obtain the same diameter as in the Kautz digraphs, and we show that MCK(d,ℓ) are d-out-regular. Finally, we compute the number of vertices of the iterated line digraphs of CK(d,ℓ).

Quasi-centers and radius related to some iterated line digraphs, proofs of several conjectures on de Bruijn and Kautz graphs

Bond (1987) and Bond et al. (1987), conjectured that a quasi-center in an undirected de Bruijn graph UB(d,D) has cardinality at least d−1, and that a quasi-center in an undirected Kautz graph UK(d,D) has cardinality at least d. They proved that for d≥3, the radii of UB(d,D) and UK(d,D) are both equals to D, and conjectured also that the radii of UB(2,D) and UK(2,D) are respectively D−1 and D. In this paper we give results in a more general context which validate these conjectures (excepting that asserting that the radius of UB(2,D) is D−1), and give simplified proofs of the cited results.

On the super‐restricted arc‐connectivity of s ‐geodetic digraphs

AbstractFor a strongly connected digraph D the restricted arc‐connectivity λ′(D) is defined as the minimum cardinality of an arc‐cut over all arc‐cuts S satisfying that D ‐ S has a non‐trivial strong component D1 such that D ‐ V (D1) contains an arc. In this paper we prove that every digraph on at least 4 vertices and of minimum degree at least 2 is λ′ ‐connected and λ′(D) ≤ξ′(D), where ξ′(D) is the minimum arc‐degree of D. Also in this paper we introduce the concept of super‐ λ′ digraphs and provide a sufficient condition for an s ‐geodetic digraph to be super‐ λ′. Further, we show that the h ‐iterated line digraph Lh(D) of an s ‐geodetic digraph is super‐ λ′ for a particular h. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

Kernels and partial line digraphs

Let D = ( V , A ) be a digraph with minimum in-degree at least 1 and girth at least l + 1 , where l ≥ 1 . In this work, the following result is proved: a digraph D has a ( k , l ) -kernel if and only if its partial line digraph L D does, where 1 ≤ l < k . As a consequence, the h -iterated line digraph L h ( D ) is shown to have a kernel if and only if D has a kernel.

Independence number of iterated line digraphs

In this paper, we deal with the independence number of iterated line digraphs of a regular digraph G. We give pertinent lower bounds and give an asymptotic estimation of the ratio of the number of vertices of a largest independent set of the n th iterated line digraph of G.

Diameter vulnerability of iterated line digraphs in terms of the girth

Iterated line digraphs arise naturally in designing fault tolerant systems. Diameter vulnerability measures the increase in diameter of a digraph when some of its vertices or arcs fail. Thus, the study of diameter vulnerability is a suitable approach to the fault tolerance of a network. In this article we present some upper bounds for diameter vulnerability of iterated line digraphs LkG. Our bounds depend basically on the girth of the digraph G and on the number of iterations k. These bounds generalize some previous results on diameter vulnerability of line digraphs. Also, we apply our results to several important families of line digraphs such as Kautz digraphs and deBruijn generalized cycles, which contain deBruijn digraphs, the Reddy-Pradhan-Kuhl digraphs, and the butterflies. Our bounds allow us to obtain improvements in known results on diameter vulnerability for all these families. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 49-54 2005

Diameter, short paths and superconnectivity in digraphs

A connected digraph is said to be superconnected if it is maximally connected and every minimum disconnecting set F consists of the vertices adjacent to or from a given vertex not belonging to F. Let δ be the minimum degree of the digraph and π be a positive integer such that π ⩽ ⌊ δ / 2 ⌋ when δ ⩾ 7 , or π ⩽ ⌊ ( δ - 2 ) / 2 ⌋ for δ ⩾ 5 . We prove that G is maximally connected or has a good superconnectivity if the diameter D ⩽ 2 ℓ π - 2 and ℓ 0 ⩾ 2 , where ℓ π is a generalization of the semigirth ℓ 0 introduced by Fàbrega and Fiol (J. Graph Theory 13(6) (1989) 657). We also show that G is maximally connected if π ⩽ ⌊ ( δ - 1 ) / 2 ⌋ and 3 ⩽ δ ⩽ 6 . In the edge case, it is enough that D ⩽ 2 ℓ π - 1 . Finally, the obtained results are applied to the iterated line digraphs.

Diameter vulnerability of GC graphs

Concern over fault tolerance in the design of interconnection networks has stimulated interest in finding large graphs with maximum degree Δ and diameter D such that the subgraphs obtained by deleting any set of s vertices have diameter at most D′, this value being close to D or even equal to it. This is the so-called ( Δ, D, D′, s)-problem. The purpose of this work has been to study this problem for s=1 on some families of generalized compound graphs. These graphs were designed by Gómez (Ars Combin. 29-B (1990) 33) as a contribution to the ( Δ, D)-problem, that is, to the construction of graphs having maximum degree Δ, diameter D and an order large enough. When approaching the mentioned problem in these graphs, we realized that each of them could be redefined as a compound graph, the main graph being the underlying graph of a certain iterated line digraph. In fact, this new characterization has been the key point to prove in a suitable way that the graphs belonging to these families are solutions to the ( Δ, D, D+1,1)-problem.

Heredity of the index of convergence of the line digraph

Let D be a digraph, and let L n ( D) and k( D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k( L n ( D))= k( D)=0 if every connected component of D is a directed cycle. (2) k( L n ( D))= k( D)+ n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that if D has no sources or no sinks then k( D)⩽ k( L( D))⩽ k( D)+1.

On Moore bipartite digraphs

AbstractIn 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 out‐degrees (d1, d2) of its partite sets of vertices. It has been proved that, when d1d2 &gt; 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 ≤ k ≤ 4. This paper deals with the problem of their enumeration. In this context, using the theory of circulant matrices and the so‐called De Bruijn near‐factorizations of cyclic groups, we present some new constructions of Moore bipartite digraphs of diameter three and composite out‐degrees. By applying the iterated line digraph technique, such constructions also provide new families of dense bipartite digraphs with arbitrary diameter. Moreover, we show that the line digraph structure is inherent in any Moore bipartite digraph G of diameter k = 4, which means that G = L G′, where G′ is a Moore bipartite digraph of diameter k = 3. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 171–187, 2003

Super link-connectivity of iterated line digraphs

Many interconnection networks can be constructed with line digraph iterations. A digraph has super link-connectivity d if it has link-connectivity d and every link-cut of cardinality d consists of either all out-links coming from a node, or all in-links ending at a node, excluding loop. In this paper, we show that the link-digraph iteration preserves super link-connectivity.