Dominating number of distance two of corona product of graphs

Dominating set S in graphG = (V,E) is a subset of V (G) such that every vertex ofGwhich is not element of S are connected and have distance one to S. Minimum cardinality among dominating sets in a graph G is called dominating number of graph G and denoted by γ(G). While dominating set of distance two which denoted by S2 is a subset of V (G) such that every vertex of G which is not element of S are connected and have maximum distance two to S2. Dominating number of distance two γ2(G) is minimum cardinality of dominating set of distance two S2. The corona G H of two graphs G and H where G has p vertices and q edges is defined as the graph G obtained by taking one copy of G and p copies of H , and then joining by an edge the i− th vertex of G to every vertex in the i − th copy of H . In this paper, we determine the dominating number of distance two of paths and cycles. We also determine the dominating number of distance two of corona product of path and any graphs as well as cycle and any graphs.


Introduction
Dominating set S in a graph G = (V, E) is a subset of V (G) such that every vertex of G which is not element of S is connected and have distance one to S. The minimum cardinality among dominating sets in a graph G is called dominating number of a graph G and denoted by γ(G).Therefore, dominating number very closely related to dominating set.
The research on dominating set in a graph have been started in 1850 when the European chess players want to solve "dominating queens" problems.In this problem, dominating set is used to determine number of queens such that every square on a standard 8 × 8 chessboard is either occupied by a queen or can be directly attacked by a queen [9].Research on dominating set and dominating number of a graph have been conducted such as c-dominating sets for families of graphs [9], dominating cartesian product of cycles [8], and total domination number of grid graph [4].
There are some applications of dominating set and dominating number of graph, i.e finding the school bus route and numbers of the bus stop such that students should not go too far from their home [7], finding number of fire engine that should be put in crossroads to cover all the region in case of fire and emergency first aid stations.Suppose that a natural disaster has struck some region consisting of many small villages.The vertices of a graph represent the villages in the region.An edge joining two vertices indicates that an emergency first aid station set up in one of the corresponding villages can also serve the other one.Then, a minimum dominating set of the graph would prescribe a way of serving the entire region with a minimum number of first aid stations [5].But if the number of the emergency first aid stations is limited, it can be minimized by assuming one emergency first aid stations can dominate two nearest villages.This motivate us to define dominating number of distance two.Graphs that will be considered are path and cycle which are operated by corona product with any graphs.The corona G H of two graphs G and H where G has p vertices and q edge is defined as the graph G obtained by taking one copy of G and p copies of H, and then joining by an edge the i − th point of G to every point in the i − th copy of H [6].
Dominating set S in a graph G = (V, E) is a subset of V (G) such that each vertex of G which is not an element of S is connected and have distance one to S. The minimum cardinality among dominating sets in a graph G is called dominating number of a graph G and denoted by γ(G) [7].While dominating set of distance two S 2 is a subset of V (G) such that each vertex G which is not an element of S 2 have distance less than or equals two to S 2 .Dominating number of distance two, is denoted by γ 2 (G), is the minimum cardinality of dominating set of distance two. Figure 1 shows the minimum dominating sets of distance one and distance two, respectively.In Figure 1.(a) vertices v 2 and v 5 represent the vertices which are the element of the minimum dominating set of distance one, while vertex u 7 in figure 1 (b) represents the vertex which is the element of the minimum dominating set of distance two.

Main Result
We begin this section with the following theorem regarding to the dominating number of distance two of path and cycle.
For dominating number of distance two of corona product graphs will be begun with the following observations regarding to the relationship between a graph G with the diameter at most 2 and its dominating number of distance two.Since the diameter of graph G is at most 2, then the distance of any two vertices in G is at most 2. By taking any v i ∈ V (G) as vertex element of dominating set of distance two S 2 (G), it implies that d(v i , V (G)) ≤ 2. Therefore, γ 2 (G) = 1.

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Dominating number of distance two of corona product of graphs | R. Umilasari and Darmaji Some graphs which have diameter less than or equals two such as Complete K n , Wheel W n , Fan F n , Friendship W m 2 , and Star S n .Thus, Observation 2.2.Let G and H be connected graphs of order m and n, respectively.If diameter of G is one Because G m and H n are corona product graphs, then Graph with the diameter equals one for example Complete K n .Then γ 2 (K n H) = 1, where H is any graph.
We now present the dominating number of distance two of corona product of path with any graph and cycle with any graph as follows.
Theorem 2.3.Let P m G n be a corona product of path P m and any graphs G n .Then dominating number of distance two The cases below show three posibilities of minimum number of vertices which are the element of dominating number of distance two on P m G n .
Case 1: S 2 ∈ V (G i ) For every v i,j element of V (G i ), v i,j can dominate at most n + 3 vertices.Then the maximal number of vertices which are the element of dominating number of distance two is mn+m n+3 , thus Case 2: S 2 ∈ V (P m ) For every v i element of V (P m ), v i can dominate at most 3n + 5 vertices.Then the maximal number of vertices which are the element of dominating number of distance two is mn+m 3n+5 , thus For every v i element of V (P m ) and v i,j element of V (G i ), then two vertices can dominate at most By three cases above mn+m 3n+5 ≤ mn+m n+4 ≤ mn+m n+3 , then we can take V (P m ) as the vertex element of S 2 .Because mn+m 3n+5 = m(n+1) 3(n+1)+2 < m(n+1) 3(n+1) = m 3 , then the interval or distance of every vertex element of S 2 is equal 3.Because |S 2 | must be integer number, then minimal number of vertices which be the element of dominating number of distance two of For every u i,j element of V (H i ), vertex u i,j can dominate at most m + 3 vertices.Then the maximal number of vertices which are the element of dominating number of distance two is nm+n m+3 , then For every u i elemen V (C n ), vertex u i can dominate at most 3m + 5 vertices.Then the maximal number of vertices which are the element of dominating number of distance two is nm+n 3m+5 , then For every u i element of V (C n ) and u i,j element of V (H i ), then two vertices can dominate at most (m + 3) + (3m + 5) vertices, thus it implies |S 2 | ≤ 2(nm+n) (m+3)+(3m+5) = nm+n m+4 .
From case 1 up to 3, nm+n 3m+5 ≤ nm+n m+4 ≤ nm+n m+3 .Then we can take the minimal number of S 2 are vertex set of Cycle C n .Because nm+n 3m+5 = n(m+1) 3(m+1)+2 < n(m+1) 3(m+1) = n 3 , then the interval or distance of every vertex element of S 2 is equal 3. We know that |S 2 | must be integer number and distance of every vertex element of S 2 is equal 3, then minimal number of vertices which are the element of dominating number of distance two of V (C n H m ) is n 3 , thus γ 2 (C n H m ) ≤ n 3 .
Therefore, to show that n 3 is minimal dominating number, we can take |S 2 | = n 3 − 1.And it makes the maximal number of vertices which are the element of S 2 is

Next, we show that m 3 3 − 1 = mn + m − n − 1 < 3 − 1 and m 3 is 3 . 2 . 4 .
is the minimal number element of S 2 .Let |S 2 | = m 3 − 1, then the maximal number of vertices which are the element of S 2 is m mn + m.It can conclude that not all vertices can be dominated.Therefore,|S 2 | = m minimum dominating number of P m G n .Thus it is proven that γ 2 (P m G n ) = mTheorem Let C n H m be a corona product of cycle C n and any graphs H m .Then dominating number of distance two γ 2