For a non-trivial connected graph G, a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the ...

A set S of vertices of a connected graph G is a geodetic set if every vertex of G lies on an x−y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set ofG is the geodetic number ofG, denoted by g(G). A set S of vertices of a graph G is an edge geodetic set if every edge of G lies on an x − y geodesic for some elements x and y in S. The minimum cardinality of an edge...

For a non-trivial connected graph G, a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the ...

Let G be an undirected graph with vertex and edge sets V (G) E(G), respectively. A subset S of vertices is a geodetic hop dominating set if it both set. The domination number G, γhg(G), the minimum cardinality among all in G. Geodetic resulting from some binary operations have been characterized. These characterizations used to determine tight bounds for each graphs considered.

For a nontrivial connected graph G = (V (G), E(G)), a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a speci...

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...

For a connected graph G = (V,E), a set S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining some pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S i...

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 ...