Minimal chordal sense of direction and circulant graphs

A sense of direction is an edge labeling on graphs that follows a globally consistent scheme and is known to considerably reduce the complexity of several distributed problems. In this paper, we study a particular instance of sense of direction, called a chordal sense of direction (CSD). In special, we identify the class of k-regular graphs that admit a CSD with exactly k labels (a minimal CSD). We prove that connected graphs in this class are Hamiltonian and that the class is equivalent to that of circulant graphs, presenting an efficient (polynomial-time) way of recognizing it when the graphs' degree k is fixed

Mathematical Properties on the Hyperbolicity of Interval Graphs

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1.This paper was supported in part by a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México and by two grants from the Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain. We would like to thank the referees for their careful reading of the manuscript and several useful comments that have helped us to improve the presentation of the paper

Structural Controllability of Multi-Agent Systems Subject to Partial Failure

Formation control of multi-agent systems has emerged as a topic of major interest during the last decade, and has been studied from various perspectives using different approaches. This work considers the structural controllability of multi-agent systems with leader-follower architecture. To this end, graphical conditions are first obtained based on the information flow graph of the system. Then, the notions of p-link, q-agent, and joint-(p,q) controllability are introduced as quantitative measures for the controllability of the system subject to failure in communication links or/and agents. Necessary and sufficient conditions for the system to remain structurally controllable in the case of the failure of some of the communication links or/and loss of some agents are derived in terms of the topology of the information flow graph. Moreover, a polynomial-time algorithm for determining the maximum number of failed communication links under which the system remains structurally controllable is presented. The proposed algorithm is analogously extended to the case of loss of agents, using the node-duplication technique. The above results are subsequently extended to the multiple-leader case, i.e., when more than one agent can act as the leader. Then, leader localization problem is investigated, where it is desired to achieve p-link or q-agent controllability in a multi-agent system. This problem is concerned with finding a minimal set of agents whose selection as leaders results in a p-link or q-agent controllable system. Polynomial-time algorithms to find such minimal sets for both undirected and directed information flow graphs are presented

On Motzkin’s problem in the circle group

The version of record is available online at: https://doi.org/10.1134/S0081543821040039Given a subset D of the interval (0,1), if a Borel set A¿[0,1) contains no pair of elements whose difference modulo 1 is in D, then how large can the Lebesgue measure of A be? This is the analogue in the circle group of a well-known problem of Motzkin, originally posed for sets of integers. We make a first treatment of this circle-group analogue, for finite sets D of missing differences, using techniques from ergodic theory, graph theory and the geometry of numbers. Our results include an exact solution when D has two elements at least one of which is irrational. When every element of D is rational, the problem is equivalent to estimating the independence ratio of a circulant graph. In the case of two rational elements, we give an estimate for this ratio in terms of the odd girth of the graph, which is asymptotically sharp and also recovers the classical solution of Cantor and Gordon to Motzkin’s original problem for two missing differences.Peer ReviewedPostprint (published version