Geodetic numbers of graphs and digraphs have been investigated in the literature recently. The main purpose of this paper is to study the geodetic spectrum of a graph. For any two vertices u and v in an oriented graph D, a u–v geodesic is a shortest directed path from u to v. Let I (u, v) denote the set of all vertices lying on a u–v geodesic. For a vertex subset A, let I (A) denote the union o...

In this paper we study monophonic sets in a connected graph G. First, we present a realization theorem proving, that there is no general relationship between monophonic and geodetic hull sets. Second, we study the contour of a graph, introduced by Cáceres and alt. [2] as a generalization of the set of extreme vertices where the authors proved that the contour of a graph is a g-hull set; in this...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

Let G : (V, E, ω) be a finite, connected, weighted graph without loops and multiple edges. In a weighted graph each arc is assigned a weight by the weight function ω: E    . A u−v path P in G is called a weighted u−v geodesic if the weighted distance between u and v is calculated along P. The strength of a path is the minimum weight of its arcs, and length of a path is the number of edges in...

Let tt : (V, E, W ) be a finite, connected, weighted graph without loops and multiple edges. In a weighted graph each arc is assigned a weight by the weight function W : E +. A u v path P in tt is called a weighted u v geodesic if the weighted distance between u and v is calculated along P . The strength of a path is the minimum weight of its arcs, and length of a path is the number of edges in...