Convexity invariants like Caratheodory, Helly and Radon numbers are computed for triangle path convexity in graphs. Unlike minimal path convexities, the Helly and Radon numbers behave almost uniformly for triangle path convexity.
A 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… (More)
The induced path transit function J (u; v) in a graph consists of the set of all vertices lying on any induced path between the vertices u and v. A transit function J satisÿes monotone axiom if x; y ∈ J (u; v) implies J (x; y) ⊆ J (u; v). A transit function J is said to satisfy the Peano axiom if, for any u; v; w ∈ V; x ∈ J (v; w), y ∈ J (u; x), there is a… (More)
Geodesic convex sets, Steiner convex sets, and induced path convex sets of lexicographic products of graphs are characterized. The geodesic case in particular rectifies [3, Theorem 3.1].
A profile on a graph G is any nonempty multiset whose elements are ver-tices from G. The corresponding remoteness function associates to each vertex x ∈ V (G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary… (More)
This Report is a preprint. It is not to be considered a formal publication in any way. It will be submitted elsewhere. Abstract An antimedian of a profile π = (x 1 , x 2 ,. .. , x k) of vertices of a graph G is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on G and… (More)
The geodesic and induced path transit functions are the two well-studied interval functions in graphs. Two important transit functions related to the geodesic and induced path functions are the triangle path transit functions which consist of all vertices on all u, v-shortest (induced) paths or all vertices adjacent to two adjacent vertices on all u,… (More)
The median (antimedian) set of a profile π = (u 1 ,. .. , u k) of vertices of a graph G is the set of vertices x that minimize (maximize) the remoteness i d(x, u i). Two algorithms for median graphs G of complexity O(n idim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles… (More)
A graph G is strongly distance-balanced if for every edge uv of G and every i ≥ 0 the number of vertices x with d(x, u) = d(x, v)−1 = i equals the number of vertices y with d(y, v) = d(y, u) − 1 = i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that… (More)