Let G= (V, E) be a graph. A subset S of V is a dominating set of G if every vertex in V \ S is adjacent to a vertex in S. A dominating set S is called a SECURE dominating set if for each v  Î V \ S there exists u 2 S such that v is adjacent to u and S1= (S \ {u})  È {v} is a dominating set. If further the vertex u Î S is unique, then S is called a PERFECT SECURE dominating set.The minimum cardinality of a PERFECT SECURE dominating set of G is called the PERFECT SECURE DOMINATION number of G and is denoted by gps (G). In this paper we initiate a study of this parameter and present several basic results.

A SECURE dominating set S ⊆ V is a dominating set of G satisfying the condition that for each u ∈ V \ S, there exists a vertex v ∈ N(u) ∩ S such that (S \ {v}) S {u} is a dominating set of G. The minimum cardinality of a SECURE dominating set of G is called the SECURE DOMINATION number of G, γs(G). In this paper, we obtain the SECURE DOMINATION number of generalized thorn paths, thorn graphs, and some special graph classes like thorn rod, thorn star and Kragujevac trees, where the generalized thorn paths are important in the study of chemical compounds.

Let G = (V, E) be a connected graph. A monophonic dominating set M is said to be a SECURE monophonic dominating set Sm (abbreviated as SMD set) of G if for each v∈V \M there exists u∈M such that v is adjacent to u and Sm = {M \(u)} ∪{v} is a monophonic dominating set. The minimum cardinality of a SECURE monophonic dominating set of G is the SECURE monophonic DOMINATION number of G and is denoted by γsm(G). In this paper, we investigate the SECURE monophonic DOMINATION number of subdivision of graphs such as subdivision of Path graph S(Pn), subdivision of Cycle graph S(Cn), subdivision of Star graph S(K1,n-1), subdivision Bistar graph S(Bm,n) and subdivision of Y-tree graph S(Yn+1).

A PERFECT Roman dominating function (PRDF) on a graph $G$ is a function $ f:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$ . The weight of a PRDF $f$ is the sum of the weights of the vertices under $f$ . The PERFECT Roman DOMINATION number of $G$ is the minimum weight of a PRDF in $G$ . In this paper we study algorithmic and computational complexity aspects of the minimum PERFECT Roman DOMINATION problem (MPRDP) . We first correct the proof of a result published in [Bulletin Iran . Math . Soc . 14(2020) , 342--351] , and using a similar argument , show that MPRDP is APX-hard for graphs with bounded degree 4 . We prove that the decision problem associated to MPRDP is NP-complete even when restricted to star convex bipartite graphs . Moreover , we show that MPRDP is solvable in linear time for bounded tree-width graphs . We also show that the PERFECT DOMINATION problem and PERFECT Roman DOMINATION problem are not equivalent in computational complexity aspects . Finally we propose an integer linear programming formulation for MPRDP .

A total Roman dominating function on a graph G is a function f: V (G)! f0; 1; 2g such that for every vertex v 2 V (G) with f(v) = 0 there exists a vertex u 2 V (G) adjacent to v with f(u) = 2, and the subgraph induced by the set fx 2 V (G): f(x)  1g has no isolated vertices. The total Roman DOMINATION number of G, denoted tR(G), is the minimum weight! (f) = P v2V (G) f(v) among all total Roman dominating functions f on G. It is known that tR(G)  t2(G) + (G) for any graph G with neither isolated vertex nor components isomorphic to K2, where t2(G) and (G) represent the semitotal DOMINATION number and the classical DOMINATION number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.

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}.