In this paper, we revisited the concept of secure dominating set introduced by Cockayne et al. We characterized secure dominating set in terms of the concept of external private neighborhood of a vertex. Also we give necessary and sufficient conditions for connected graphs to have secure domination number equal to 2. We give some characterizations of the concept of secure domination in the joins K 1 + G and G + H, where G and H are connected non-complete graphs and then determine their corresponding secure domination numbers.

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Given a connected graph G and two vertices u and v in V (G), I G [u, v] denotes the closed interval consisting of u, v and all vertices lying onevery pair of vertices u, v ∈ C. A subset T of a maximum convex set Sergio R. Canoy Jr. et al.S of G is called a forcing subset for S if S is the unique maximum convex set containing T . The forcing convexity number f con(S) of a maximum convex set S, is the minimum cardinality of a forcing subset for S. The forcing convexity number f con(G) of G is the smallest f con(S) among all maximum convex sets S of G. In this paper, we characterize the forcing subsets in the join and composition of graphs and determine their forcing convexity numbers. Mathematics Subject Classification: 05C69

In this paper, a quadrex algorithm for quadratic programming problems is introduced (n = 2) under linear and quadratic constraints. The quadrex algorithm considers on the behavior of the quadratic function near the origin or a translate of the origin, performs a series of translationsand orthogonal rotations to obtain the optimal solution of the objective function as well as taking considerations on the constraints of the problem. The method works provided that the eigenvalues of the matrix on quadratic form of the objective function is strictly negative, that is,Q is negative-definite. The quadrex algorithm is a parallel counterpart of the simplex algorithm for linear programming models.

This paper further investigates the cyclic group ( ) * p Z with respect to the primitive roots or generators. The simulation algorithm that determines the generators and the number of generators, g of ( ) * p Z for a prime p is illustrated using Python programming.The probability of getting a generator g of ( ) * p Z , denoted by , is generated for prime p between 0 to 3000. The scatterplot is also shown that depicts the data points on the probability of the group of units with respect to the order p -1 of for prime p between 0 to 3000. The scatterplot results reveal that the probability of getting a generator of the group of units is fluctuating within the probability range of 0.20 to 0.50, for prime p modulus from 3 to 3000. These findings suggest that the proportion of the number of generators of the group of units modulo a prime of order p -1, though fluctuating, is bounded from 20% to 50% for prime p modulus from 3 to 3000.Keywords: Group of units modulo a prime, , primitive roots or generators of , , simulation algorithm, probability of getting a generator g of .