Convexities related to path properties on graphs

AbstractA feasible family of paths in a connected graph G is a family that contains at least one path between any pair of vertices in G. Any feasible path family defines a convexity on G. Well-known instances are: the geodesics, the induced paths, and all paths. We propose a more general approach for such ‘path properties’. We survey a number of results from this perspective, and present a number of new results. We focus on the behaviour of such convexities on the Cartesian product of graphs and on the classical convexity invariants, such as the Carathéodory, Helly and Radon numbers in relation with graph invariants, such as the clique number and other graph properties

Convexities related to path properties on graphs; a unified approach

Path properties, such as 'geodesic', 'induced', 'all paths' define a convexity on a connected graph. The general notion of path property, introduced in this paper, gives rise to a comprehensive survey of results obtained by different authors for a variety of path properties, together with a number of new results. We pay special attention to convexities defined by path properties on graph products and the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants, such as clique numbers and other graph properties.

The induced path function, monotonicity and betweenness

The induced path function $J(u, v)$ of a graph consists of the set of all vertices lying on the induced paths between vertices $u$ and $v$. This function is a special instance of a transit function. The function $J$ satisfies betweenness if $w \\in J(u, v)$ implies $u \\notin J(w, v)$ and $x \\in J(u, v)$ implies $J(u, x \\subseteq J(u, v)$, and it is monotone if $x, y \\in J(u, v)$ implies $J(x, y) \\subseteq J(u, v)$. The induced path function of aconnected graph satisfying the betweenness and monotone axioms are characterized by transit axioms.betweenness;induced path;transit function;monotone;house domino;long cycle;p-graph

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

Steiner distance and convexity in graphs

We use the Steiner distance to define a convexity in the vertex set of a graph, which has a nice behavior in the well-known class of HHD-free graphs. For this graph class, we prove that any Steiner tree of a vertex set is included into the geodesical convex hull of the set, which extends the well-known fact that the Euclidean convex hull contains at least one Steiner tree for any planar point set. We also characterize the graph class where Steiner convexity becomes a convex geometry, and provide a vertex set that allows us to rebuild any convex set, using convex hull operation, in any graph

Transit functions on graphs (and posets)

The notion of transit function is introduced to present a unifying approachfor results and ideas on intervals, convexities and betweenness in graphs andposets. Prime examples of such transit functions are the interval function I andthe induced path function J of a connected graph. Another transit function isthe all-paths function. New transit functions are introduced, such as the cutvertextransit function and the longest path function. The main idea of transitfunctions is that of ‘transferring’ problems and ideas of one transit functionto the other. For instance, a result on the interval function I might suggestsimilar problems for the induced path function J. Examples are given of howfruitful this transfer can be. A list of Prototype Problems and Questions forthis transferring process is given, which suggests many new questions and openproblems.graph theory;betweenness;block graph;convexity;distance in graphs;interval function;path function;induced path;paths and cycles;transit function;types of graphs

On the hull and interval numbers of oriented graphs

In this work, for a given oriented graph $D$, we study its interval and hull numbers, denoted by ${in}(D)$ and ${hn}(D)$, respectively, in the geodetic, ${P_3}$ and ${P_3^*}$ convexities. This last one, we believe to be formally defined and first studied in this paper, although its undirected version is well-known in the literature. Concerning bounds, for a strongly oriented graph $D$, we prove that ${hn_g}(D)\leq m(D)-n(D)+2$ and that there is a strongly oriented graph such that ${hn_g}(D) = m(D)-n(D)$. We also determine exact values for the hull numbers in these three convexities for tournaments, which imply polynomial-time algorithms to compute them. These results allows us to deduce polynomial-time algorithms to compute ${hn_{P_3}}(D)$ when the underlying graph of $D$ is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether ${in_g}(D)\leq k$ or ${hn_g}(D)\leq k$ is NP-hard or W[i]-hard parameterized by $k$, for some $i\in\mathbb{Z_+^*}$, then the same holds even if the underlying graph of $D$ is bipartite. Next, we prove that deciding whether ${hn_{P_3}}(D)\leq k$ or ${hn_{P_3^*}}(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is bipartite; that deciding whether ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ is NP-complete, even if $D$ has no directed cycles and the underlying graph of $D$ is a chordal bipartite graph; and that deciding whether ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is split. We also argue that the interval and hull numbers in the oriented $P_3$ and $P_3^*$ convexities can be computed in polynomial time for graphs of bounded tree-width by using Courcelle's theorem

The induced path function, monotonicity and betweenness

The induced path function $J(u, v)$ of a graph consists of the set of all vertices lying on the induced paths between vertices $u$ and $v$. This function is a special instance of a transit function. The function $J$ satisfies betweenness if $w \\in J(u, v)$ implies $u \\notin J(w, v)$ and $x \\in J(u, v)$ implies $J(u, x \\subseteq J(u, v)$, and it is monotone if $x, y \\in J(u, v)$ implies $J(x, y) \\subseteq J(u, v)$. The induced path function of a connected graph satisfying the betweenness and monotone axioms are characterized by transit axioms