Connectivity of consecutive-d digraphs

AbstractThe concept of consecutive-d digraph is proposed by Du, Hsu and Hwang. It generalizes the class of de Bruijin digraphs, the class of Imase-Itoh digraphs and the class of generalized de Bruijin graphs. We modify consecutive-d digraphs by connecting nodes with a loop into a circuit and deleting all loops. The result in this paper shows that the link-connectivity or the connectivity of modified consecutive-d digraphs get better

On hardware for generating routes in Kautz digraphs

In this paper we present a hardware implementation of an algorithm for generating node disjoint routes in a Kautz network. Kautz networks are based on a family of digraphs described by W.H. Kautz[Kautz 68]. A Kautz network with in-degree and out-degree d has N = dk + dk¿1 nodes (for any cardinals d, k>0). The diameter is at most k, the degree is fixed and independent of the network size. Moreover, it is fault-tolerant, the connectivity is d and the mapping of standard computation graphs such as a linear array, a ring and a tree on a Kautz network is straightforward.\ud The network has a simple routing mechanism, even when nodes or links are faulty. Imase et al. [Imase 86] showed the existence of d node disjoint paths between any pair of vertices. In Smit et al. [Smit 91] an algorithm is described that generates d node disjoint routes between two arbitrary nodes in the network. In this paper we present a simple and fast hardware implementation of this algorithm. It can be realized with standard components (Field Programmable Gate Arrays)

Sizing the length of complex networks

Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a small-world network and how does it compare to others' has remained a difficult challenge to answer. Here we establish a new synoptic representation that allows for a complete and accurate interpretation of the pathlength (and efficiency) of complex networks. We frame every network individually, based on how its length deviates from the shortest and the longest values it could possibly take. For that, we first had to uncover the upper and the lower limits for the pathlength and efficiency, which indeed depend on the specific number of nodes and links. These limits are given by families of singular configurations that we name as ultra-short and ultra-long networks. The representation here introduced frees network comparison from the need to rely on the choice of reference graph models (e.g., random graphs and ring lattices), a common practice that is prone to yield biased interpretations as we show. Application to empirical examples of three categories (neural, social and transportation) evidences that, while most real networks display a pathlength comparable to that of random graphs, when contrasted against the absolute boundaries, only the cortical connectomes prove to be ultra-short

Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$such that$uv\in A(G)$implies$f(u)f(v)\in A(H)$. If, moreover, each vertex$u \in V(G)$is associated with costs$c_i(u), i \in V(H)$, then the cost of a homomorphism$f$is$\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph$H$, the minimum cost homomorphism problem for$H$, denoted MinHOM($H$), can be formulated as follows: Given an input digraph$G$, together with costs$c_i(u)$,$u\in V(G)$,$i\in V(H)$, decide whether there exists a homomorphism of$G$to$H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes