Geodetic sets in graphs are briefly surveyed. After an overview of earlier results, we concentrate on recent studies of the geodetic number and related invariants in graphs. Geodetic sets in Cartesian products of graphs and in median graphs are considered in more detail. Algorithmic issues and relations with several other concepts, arising from various convex and interval structures in graphs, ...

Let u, v ∈ V be two vertices of a connected graph G. The vertex v is said to be a boundary vertex of u if no neighbor of v is further away from u than v. The boundary of a graph is the set of all its boundary vertices. In this work, we present a number of properties of the boundary of a graph under different points of view: (1) a realization theorem involving different types of boundary vertex ...

We show that for every integer k ≥ 2 and every k graphs G1, G2, . . . , Gk, there exists a hull graph with k hull vertices v1, v2, . . . , vk such that link L(vi) = Gi for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a, b of integers with a ≥ 2 and b ≥...

Let G be a connected graph of order n and S ⊆ V (G). A closed geodetic cover Gis path-induced dominating set if subgraph <S> has Hamiltonianpath is G. The minimum cardinality geodeticdominating called domination number This study presentsthe characterization the sets some common graphsand edge corona two graphs. numbers thesegraphs are also determined.

Given a graph G and a subset W ⊆ V (G), a Steiner W -tree is a tree of minimum order that contains all of W . Let S(W ) denote the set of all vertices in G that lie on some Steiner W -tree; we call S(W ) the Steiner interval of W . If S(W ) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G. Given two vertices u, v in G, a short...

Let G be a connected graph. For two vertices u and v in G, a u–v geodesic is any shortest path joining u and v. The closed geodetic interval IG[u, v] consists of all vertices of G lying on any u–v geodesic. For S ⊆ V (G), S is a geodetic set in G if ⋃ u,v∈S IG[u, v] = V (G). Vertices u and v of G are neighbors if u and v are adjacent. The closed neighborhood NG[v] of vertex v consists of v and ...

For any two vertices u and v in a graph G (digraph D, respectively), a u-v geodesic is a shortest path between u and v (from u to v, respectively). Let I(u, v) (ID(u, v), respectively) denote the set of all vertices lying on a u-v geodesic. For a vertex subset S, let IG(S) (ID(S), respectively) denote the union of all IG(u, v) (ID(u, v), respectively) for u, v ∈ S. The geodetic number g(G) (g(D...