Let F1, F2, . . . , Fk be graphs with the same vertex set V . A subset S ⊆ V is a factor dominating set if in every Fi every vertex not in S is adjacent to a vertex in S, and a factor total dominating set if in every Fi every vertex in V is adjacent to a vertex in S. The cardinality of a smallest such set is the factor (total) domination number. In this note we investigate bounds on the factor ...

a r t i c l e i n f o a b s t r a c t For a fixed positive integer k, a k-tuple total dominating set of a graph G = (V , E) is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k. In minimum k-tuple total dominating set problem (Min k-Tuple Total Dom Set), it is required to find a k-tuple total dominating set of minimum cardinality and Decide Min k-Tuple ...

Let G = (V,E) be a graph. A set S ⊆ V (G) is a total restrained dominating set if every vertex of G is adjacent to a vertex in S and every vertex of V (G)\S is adjacent to a vertex in V (G)\S. The total restrained domination number of G, denoted by γtr(G), is the smallest cardinality of a total restrained dominating set of G. In this paper we continue the study of total restrained domination in...

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γt(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V (G) ...

Let G be a graph with vertex set V . A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V \D has a neighbor in V \D. The minimum cardinality of a total restrained dominating set of G is called the total restrained domination number of G, and is denoted by γtr(G). In this paper, we prove that if G is a connected graph of order n ≥ 4...

Given a semigraph, we can construct graphs Sa, Sca, Se and S1e. In the same pattern, we construct bipartite graphs CA(S), A(S), VE(S), CA(S) and A(S). We find the equality of domination parameters in the bipartite graphs constructed with the domination and total domination parameters of the graphs Sa and Sca. We introduce the domination and independence parameters for the bipartite semigraph. W...

A edge 2-rainbow dominating function (E2RDF) of a graph G is a ‎function f from the edge set E(G) to the set of all subsets‎ ‎of the set {1,2} such that for any edge.......................

A total outer-independent dominating set of a graph G = (V (G), E(G)) is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \D is independent. The total outer-independent domination number of a graph G, denoted by γ t (G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every tree T of order n ≥ 4, with l le...

Nine variations of the concept of domination in a simple graph are identified as fundamental domination concepts, and a unified approach is introduced for studying them. For each variation, the minimum cardinality of a subset of dominating elements is the corresponding fundamental domination number. It is observed that, for each nontrivial connected graph, at most five of these nine numbers can...

A locating-total dominating set (LTDS) S of a graph G is a total dominating set S of G such that for every two vertices u and v in V(G) − S, N(u)∩S ≠ N(v)∩S. The locating-total domination number ( ) l t G  is the minimum cardinality of a LTDS of G. A LTDS of cardinality ( ) l t G  we call a ( ) l t G  -set. In this paper, we determine the locating-total domination number for the special clas...