In this paper, we study both concepts of geodetic dominating and edge geodetic dominating sets and derive some tight upper bounds on the edge geodetic and the edge geodetic domination numbers. We also obtain attainable upper bounds on the maximum number of elements in a partition of a vertex set of a connected graph into geodetic sets, edge geodetic sets, geodetic dominating sets and edge geodetic dominating sets, respectively.

A subset S of vertices in a graph G is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbor in D. A geodetic dominating set S is both a geodetic and a dominating set. The geodetic (domination, geodetic domination) number g(G) (g(G), gg (G)) of G is the minimum cardinality among all geodetic (dominating, geodetic dominating) sets in G. In this paper, we show that if a triangle free graph G has minimum degree at least 2 and g (G) =2, then gg(G) = g(G). It is shown, for every nontrivial connected graph G with g (G) =2 and diam (G)>3, that gg (G)>g (G). The lower bound for the geodetic domination number of Cartesian product graphs is proved. geodetic domination number of product of cycles (paths) are determined. In this work, we also determine some bounds and exact values of the geodetic domination number of strong product of graphs.

In this paper, we introduced the concept of edge geodetic sequences in graph and its generating function. Some general properties satisfied by this concept are studied. It is shown that for every generating function$$ G(x)=\sum_{i=1}^{\infty} {a}^{i-1}{x^{i-1}} \quad a\in N-\left\lbrace 1\right\rbrace,$$there exists a recurrence graph $G$ with edge geodetic decomposition $\pi=\{G_{1},G_{2},\ldots ,G_{n}\ldots\}$.

Let G = (V; E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection  of edge-disjoint subgraphs G1; G2; : : :; Gn of G such that every edge of G belongs to exactly one Gi; (1  i  n). The decomposition  = fG1; G2; : : :; Gng of a connected graph G is said to be a distinct edge geodetic decomposition if g1(Gi) 6= g1(Gj ); (1  i 6= j  n). The maximum cardinality of  is called the distinct edge geodetic decomposition number of G and is denoted by  dg1 (G), where g1(G) is the edge geodetic number of G. Some general properties sat-is ed by this concept are studied. Connected graphs of  dg1 (G)  2 are characterized and connected graphs of order p with  dg1 (G) = p 2 are characterized.

In this paper, we investigate domination number as well as signed domination numbers of Cay(G: S) for all cyclic group G of order n, where n ϵ {pm, pq} and S = {k < n: gcd(k, n) = 1}. We also introduce some families of connected regular graphs 􀀀 such that S (􀀀 ) ϵ {2, 3, 4, 5}.

An edge $e$ of a simple graph $G=(V_{G},E_{G})$ is said to ev-dominate a vertex $v\in V_{G}$ if $e$ is incident with $v$ or $e$ is incident with a vertex adjacent to $v$. A subset $D\subseteq E_{G}$ is an edge-vertex dominating set (or an evd-set for short) of $G$ if every vertex of $G$ is ev-dominated by an edge of $D$. The edge-vertex domination number of $G$ is the minimum cardinality of an evd-set of $G$. In this paper, we initiate the study of the graphs with unique minimum evd-sets that we will call UEVD-graphs. We first present some basic properties of UEVD-graphs, and then we characterize UEVD-trees by equivalent conditions as well as by a constructive method.