Daniela Ferrero

Algebraic properties of a digraph and its line digraph

Let G be a digraph, LG its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows us to relate properties of G and LG, specifically the number of cycles of a given length.

Edge-connectivity and super edge-connectivity of P 2 -path graphs

For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in G. Two vertices are connected when their union is a path or a cycle of length 3. We present lower bounds on the edge-connectivity, λ(P2(G)) of a connected graph G and give conditions for maximum connectivity. A maximally edge-connected graph is super-λ if each minimum edge cut is trivial, and it is optimum super-λ if each minimum nontrivial edge cut consists of all the edges adjacent to one edge. We give conditions on G, for P2(G) to be super-λ and optimum super-λ.

Connectivity and fault-tolerance of hyperdigraphs

Directed hypergraphs are used to model networks whose nodes are connected by directed buses. We study in this paper two parameters related to the fault-tolerance of directed bus networks: the connectivity and the fault-diameter of directed hypergraphs. Some bounds are given for those parameters. As a consequence, we obtain that de Bruijn-Kautz directed hypergraphs and, more generally, iterated line directed hypergraphs provide models for highly fault-tolerant directed bus networks.

Power domination in cylinders, tori, and generalized Petersen graphs

A set S of vertices is defined to be a power dominating set (PDS) of a graph G if every vertex and every edge in G can be monitored by the set S according to a set of rules for power system monitoring. The minimum cardinality of a PDS of G is its power domination number. In this article, we find upper bounds for the power domination number of some families of Cartesian products of graphs: the cylinders Pn□Cm for integers n ≥ 2, m ≥ 3, and the tori Cn□Cm for integers n,m ≥ 3. We apply similar techniques to present upper bounds for the power domination number of generalized Petersen graphs P(m,k). We prove those upper bounds provide the exact values of the power domination numbers if the integers m,n, and k satisfy some given relations.

Note on Power Propagation Time and Lower Bounds for the Power Domination Number

We present a counterexample to a lower bound for the power domination number given in Liao (J Comb Optim 31:725–742, 2016). We also define the power propagation time, using the power domination propagation ideas in Liao and the (zero forcing) propagation time in Hogben et al. (Discrete Appl Math 160:1994–2005, 2012).

Disjoint paths of bounded length in large generalized cycles

Abstract A generalized p-cycle is a digraph whose set of vertices is partitioned in p parts that can be ordered in such a way that a vertex is adjacent only to vertices in the next part. The families of BGC(p,d,dp) and KGC(p,d,dp · k + dp) are the largest known p-cycles for their degree and diameter. In this paper we present a lower bound for the fault-diameter of a generalized cycle. Then we calculate the wide-diameter and the fault-diameter of the families mentioned above, by constructing disjoint paths between any pair of vertices. We conclude that the values of these parameters for BGC(p,d,dp) and KGC(p,d,dp+k + dp) exceed the lower bound at most in one unit.

Power domination in honeycomb networks

Abstract Electric power networks must be continuously monitored. Such monitoring can be efficiently accomplished by placing phase measurement units (PMUs) at selected network locations. Due to the high cost of the PMUs, their number must be minimized. The power domination problem consists of finding the minimum number of PMUs needed to monitor a given electric power system. The power dominating problem is NP-hard, but closed formulas for the power domination number of certain networks, such as rectangular meshes [4] have been found. In this work we extend the results for rectangular meshes to honeycomb meshes.

New bounds on the diameter vulnerability of iterated line digraphs

Abstract Iterated line digraphs have some good properties in relation to the design of interconnection networks. The diameter vulnerability of a digraph is the maximum diameter of the subdigraphs obtained by deleting a fixed number of vertices or arcs. This parameter is related to the fault-tolerance of interconnection networks. In this work, we introduce some new parameters in order to find new bounds for the diameter vulnerability of general iterated line digraphs.

Restricted power domination and zero forcing problems

Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X. We also study the related problem of finding the smallest zero forcing set that contains a given set of vertices X. The sizes of such sets in a graph G are respectively called the restricted power domination number and restricted zero forcing number of G subject to X. We derive several tight bounds on the restricted power domination and zero forcing numbers of graphs, and relate them to other graph parameters. We also present exact and algorithmic results for computing the restricted power domination number, including integer programs for general graphs and a linear time algorithm for graphs with bounded treewidth. We also use restricted power domination to obtain a parallel algorithm for finding minimum power dominating sets in trees.

Partial line directed hypergraphs

The partial line digraph technique was introduced in [7] in order to construct digraphs with a minimum diameter, maximum connectivity, and good expandability. To find a new method to construct directed hypergraphs with a minimum diameter, we present in this paper an adaptation of that technique to directed hypergraphs. Directed hypergraphs are used as models for interconnection networks whose vertices are linked by directed buses. The connectivity and expandability of partial line directed hypergraphs are studied. Besides, we prove a conjecture by J-C. Bermond and F. Ergincan about the characterization of line directed hypergraphs.