Variant Domination Types for a Complete h-ary Tree

Graph G = (V, E) is a tool that can be used to simplify and solve network problems. Domination is a typical network problem that graph theory is well suited for. A subset of nodes in any network is called dominating if every node is contained in this subset, or is connected to a node in it via an edge. Because of the importance of domination in different areas, variant types of domination have been introduced according to the purpose they are used for. In this paper, two domination parameters the first is the restrained and the second is secure domination have been chosn. The secure domination, and some types of restrained domination in one type of trees is called complete h −ary tree(Tc,h,r) are determined.

⊆ ( ) of a graph together with any edges whose endpoints are both in this subset. Any notion or definition which is not found here could be found in (1,2). An independent vertex set of a graph G is a subset of the vertices such that no two vertices in the subset represent an edge of , (2).
In any graph the set ⊆ ( ) is a dominating set if every vertex ∈ is either an element of or is adjacent to an element of . The domination number of , denoted by ( ), is the cardinality of a minimum dominating set of , (3). Because of the importance of domination in different areas, variant types of domination have been introduced according to the purpose they are used for. The domination parameters have been formed either by putting a condition on the vertices of a dominating set , or by putting a condition on the vertices inor on both. A total dominating set of a graph is a dominating such that [ ] has no isolated vertices.st. An independent dominating set is a vertex subset which is both independent and dominating.
Various types of domination of a graph have been defined and studied by several authors, they are listed in the appendix of Haynes (4,5). For more details about parameters of domination number that depend on vertex dominating with condition on dominating set , see (6,7,8,9), and for condition on vertices in set -, see (10,11,12,13,14).
Here, variation types of the domination theme, namely that of restrained domination are studied. In a graph with a dominating set , and the following condition put on the vertices of set − , such that the open neighborhood of every vertex in [ − ] is not an empty set. If this condition is verified then is a restrained dominating set in . The number γ ( ), is the minimum cardinality of a restrained dominating set of , (15). When, is a restrained and independent then it is called an independent restrained dominating set, (16 ). The independent restrained domination number of , denoted by γ ( ), is the smallest cardinality of an independent restrained dominating set of .
The total restrained domination number of , denoted by γ ( ), is the smallest cardinality of a total restrained dominating set of (when the dominating set is restrained and has no isolated vertices in [ ]), (17). For the following condition: each ∈ − , there exists a vertex ∈ such that ∈ and ( − { }) ∪ { } is a dominating set, if the this condition is verified then is a secure dominating set of .The minimum cardinality of a secure dominating set in is the secure domination number denoted by γ ( ), (18,19). In this paper, the domination in one type of trees is studied. Let ℎ ≥ 1 , ≥ 0 be integers. The complete h − ary tree of depth ,denoted by ( ,ℎ, ), is a complete ( ) rooted tree in which every non-pendent vertex has exactly ℎ children, and the distance from the root to each pendent is exactly . The root vertex h been labeled by 0 , (as an example, see Fig. 1).
Here, various types of dominating parameter (restrained domination, total restrained domination, independence restrained domination, and secure domination) number of a complete ℎ-ary root, ℎ ≥ 2, ≥ 3 are determined.

Restrained domination in a complete -ary tree
In this section, restrained, independent restrained and total restrained domination for a complete ℎ-ary tree ,ℎ, , are determined:

Proof.
Looking for a set such that, this set contains as possible a minimum restrained dominating set. So, . Three cases are obtained.

Proof.
Looking for a set such that, this set contains as possible a minimum total restrained dominating set. Let ∪ { } be the dominating set of where, is any vertex of depth one. So, ∪ { } is a minimum total restrained dominating set in . Therefore the result is gotten.  Secure domination in a complete -ary tree Theorem 5 If = ,ℎ, is a complete ℎ-ary tree; then for ≥ 3 ,
where ={ : is a vertex of depth − 2 − 1 } in , Thus, all vertices of are dominated by set .

Every vertex in
is adjacent to some vertices in row +1 . Therefore, for each vertex in there is a vertex in − such that, the swap ( ( − { })⋃{ }) is a domination set since the vertices in +1 are dominated by the vertices of in row and the vertices of +2 in row +2 .Therefore, is the secure dominating set in . Thus, ( ) = ℎ +1 −1 ℎ 2 −1 +ℎ −1 (ℎ − 1). If there is another dominating set say , and | | < | | , then there are at least ℎ vertices of which are not dominated by set . Thus, is the minimum and it is a secure dominating set. So that, the secure domination number in this case of ,ℎ, is where ={ : is a vertex of depth − 2 } in , and . While, 2 is the same set in Case1.

Conclusion:
Domination number for some types of graph domination is calculated for a complete ℎ-ary root tree ,ℎ, . A restrained domination number for complete ℎ -ary root, ℎ ≥ 2, can be determined form ≥ 3, while the total restrained domination can be determined from ≥ 5. When, = 3 4 the complete ℎ -ary tree has no total restrained domination.
The independence restrained domination number equals to restrained domination number for the complete ℎ -ary tree, when ≡ 0( 3). When ( ≡ 1,2( 3), the graph has no independence restrained dominating set. The secure domination in a complete ℎ-ary tree can be determined from ≥ 3.