For a graph G=(V,E) with V=V(G) and E=E(G), perfect double Italian dominating function is f:V→{0,1,2,3} having the property that 3≤∑u∈NG[v]f(u)≤4, for every vertex v∈G f(v)∈{0,1}. The weight of f sum f(V)=∑v∈V(G)f(v) minimum on G domination number γdIp(G) G. We initiate study functions. check γdIp some standard graphs evaluate γdI such graphs. functions versus Roman are perused. NP-completeness...

A function f : V (G) → {−1, 0, 1} is a minus dominating function if for every vertex v ∈ V (G), ∑ u∈N [v] f(u) ≥ 1. A minus dominating function f of G is called a global minus dominating function if f is also a minus dominating function of the complement G of G. The global minus domination number γ− g (G) of G is defined as γ − g (G) = min{ ∑ v∈V (G) f(v) | f is a global minus dominating functi...

A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X − {v}) ∪ {u} is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary s...

A set D of vertices in a graph G is 2-dominating if every vertex not in D has at least two neighbors in D and locating-dominating if for every two vertices u, v not in D, the sets N(u) ∩ D and N(v) ∩ D are non-empty and different. The minimum cardinality of a 2-dominating set (locatingdominating set) is denoted by γ2(G) (γL(G)). It is known that every tree T with n ≥ 2 vertices, leaves, s suppo...

A dominating set S of a graph G is a global (strong) defensive alliance if for every vertex v ∈ S, the number of neighbors v has in S plus one is at least (greater than) the number of neighbors it has in V \ S. The dominating set S is a global (strong) offensive alliance if for every vertex v ∈ V \ S, the number of neighbors v has in S is at least (greater than) the number of neighbors it has i...

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = k n = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the conception of k-domina...

A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3} with the properties that if f(u)=0, then vertex u adjacent to at least one assigned 3 or two vertices 2, and f(u)=1, 2 3. The weight of f equals w(f)=?v?Vf(v). domination number ?dR(G) G minimum G. said be ?dR(G)=3?(G), where ?(G) We obtain sharp lower bound generalized Petersen graphs P(3k,k), we construct solutions providi...

Results of electro-thermo-elastic analysis of a functionally graded thick-walled spherical shell made of temperature dependent materials are presented in this article. All material properties are assumed temperature-dependent and also are graded along the thickness direction based on power function. Temperature dependency is accounted for all material properties including, thermal, mechanical a...

‎for any integer $kge 1$‎, ‎a minus $k$-dominating function is a‎ ‎function $f‎ : ‎v (g)rightarrow {-1,0‎, ‎1}$ satisfying $sum_{win‎‎n[v]} f(w)ge k$ for every $vin v(g)$‎, ‎where $n(v) ={u in‎‎v(g)mid uvin e(g)}$ and $n[v] =n(v)cup {v}$‎. ‎the minimum of‎‎the values of $sum_{vin v(g)}f(v)$‎, ‎taken over all minus‎‎$k$-dominating functions $f$‎, ‎is called the minus $k$-domination‎‎number and i...

A dominating set S of a graph G is a locating-dominating-set, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LDcodes and the cardinality of an LD-code is the location-domination number, λ(G). An LD-set S of a graph G is global if it is an LD-set for both G and its complem...