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

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

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

Helly theorems for 3-Steiner and 3-monophonic convexity in graphs

AbstractA family C of sets has the Helly property if any subfamily C′ whose elements are pairwise intersecting has non-empty intersection. Suppose that C is a non-empty family of subsets of a finite set V: the Helly number h(C) of C is the least positive integer n such that every n-wise intersecting subfamily of C has non-empty intersection.In this paper the Helly property of families of convex sets relative to two new graph convexities are studied. Let G be a (finite) connected graph and U a set of vertices of G. A connected subgraph with the fewest edges containing U is called a Steiner tree for U, and the collection of all vertices of G that belong to some Steiner tree for U is called the Steiner interval for U. A set S of vertices of G is g3-convex if it contains the Steiner interval for every 3-subset U of S. A subtree T of G that contains U is a minimal U-tree if every vertex of T that is not in U is a cut-vertex of the subgraph induced by V(T). The set of all vertices that belong to some minimal U-tree is called the monophonic interval for U and a set S of vertices is m3-convex if it contains the monophonic interval of every 3-subset U of S. Those graphs are characterized for which the families of g3-convex (m3-convex) sets of size at least 3 have the Helly property. A graph obtained from a complete graph by deleting a matching is called a near-clique. The maximum order of a near-clique in a graph G is called the near-clique number of G. The near-clique number of a graph is a lower bound on the Helly number for both g3-convex families and m3-convex families. For m3-convex families equality holds. For g3-convex families equality holds for chordal graphs and for distance-hereditary graphs, but the bound can be arbitrarily bad in general, even when the near-clique number is 3