Byzantine Approximate Agreement on Graphs

Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors\u27 input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures

Monophonic numbers of the join and composition of connected graphs

AbstractIn this paper, we describe the monophonic sets in the join and composition of two connected graphs. We also determine the monophonic numbers of the join of any two connected graphs and the composition G[Kn] of a connected graph G and the complete graph Kn. Lower and upper bounds are obtained for the monophonic number of the composition G[H], where G is connected and H is a connected non-complete graph

The restrained monophonic number of a graph

A set S of vertices of a connected graph G is a monophonic set of G if each vertex v of G lies on a x−y monophonic path for some x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G). A restrained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V − S has no isolated vertices. The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by mr(G). We determine bounds for it and determine the same for some special classes of graphs. Further, several interesting results and realization theorems are proved.Publisher's Versio

The general position number and the iteration time in the P3 convexity

In this paper, we investigate two graph convexity parameters: the iteration time and the general position number. Harary and Nieminem introduced in 1981 the iteration time in the geodesic convexity, but its computational complexity was still open. Manuel and Klav\v{z}ar introduced in 2018 the general position number of the geodesic convexity and proved that it is NP-hard to compute. In this paper, we extend these parameters to the P3 convexity and prove that it is NP-hard to compute them. With this, we also prove that the iteration number is NP-hard on the geodesic convexity even in graphs with diameter two. These results are the last three missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and P3 convexities

Toll convexity

A walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W is among vertices from W adjacent only to the second vertex of W, and the last vertex of W is among vertices from W adjacent only to the second-last vertex of W. In the resulting interval convexity, a set S ⊂ V(G) is toll convex if for any two non-adjacent vertices x, y ∈ S any vertex in a tolled walk between x and y is also in S. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski-Krein-Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.Facultad de Ciencias Exacta

Toll convexity

A walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W is among vertices from W adjacent only to the second vertex of W, and the last vertex of W is among vertices from W adjacent only to the second-last vertex of W. In the resulting interval convexity, a set S ⊂ V(G) is toll convex if for any two non-adjacent vertices x, y ∈ S any vertex in a tolled walk between x and y is also in S. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski-Krein-Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.Facultad de Ciencias Exacta