Block Split Subdivision Domination in Graphs

A dominating set is a split dominating set in , if the induced subgraph is disconnected in . The split domination number of is denoted by , is the minimum cardinality of a split dominating set in . In this paper, some results on were obtained interms of vertices, blocks and other different parameters of G but not the members of . Further we developed its relationship with other different domination parameters of


Introduction
All graphs, subdivision graphs and block subdivision graphs considered here are simple, finite, nontrivial, undirected and connected, as usual denotes the number of vertices, edges and blocks of a graph G respectively.In this paper, the study of domination in graphs started by C. Berge [1] and O. Ore [2].The graph theoretical terminology undefined or notations can be found in G. Chartrand and P. Zhang [3] and F. Harary [4].The domination in graphs is discussed by T. W. Haynes [5].
As usual, the minimum degree and maximum degree of a graph G are denoted by respectively.A vertex is called a cut vertex if removing it from G increases the number of components of G.For any real number denotes the greatest integer not greater than denotes the smallest integer not less than .The complement of a graph G has as its vertex set, but two vertices are adjacent in , if they are not adjacent in G.The vertex independence number is the maximum cardinality of an independence set of vertices.An edge cover of G is the set of edges that covers all the vertices.The edge covering number is minimum cardinality of an edge cover.The edge independence number is the minimum cardinality of an independent set of edges.A set of vertices is a dominating set, if every vertex in is adjacent to some vertex in D. The dominance number is the minimum cardinality of a dominating set in G [6].A dominating set is a cototal dominating set, if the induced sub graph has no isolated vertices.The cototal domination number is the minimum cardinality of a cototal dominating set [6].A dominating set D of a graph G is a split dominating set, if the induced subgraph is disconnected.The split domination number of a graph G is the minimum cardinality of a split dominating set.This concept was introduced by Kulli [6,7].A dominating set D is a connected dominating set, if the induced sub graph is connected.The connected domination number of a connected graph G is the minimum cardinality of a connected dominating set [8].The dominating set D is a total dominating set, if the induced subgraph has no isolated vertices.The total domination number of a graph G is the minimum cardinality of a total dominating set [9].
The following figures illustrate the formation of of a graph G. Blocks of G = n = 3 The domination of block split subdivision domination in graph is denoted by .In this paper some results on were obtained interms of vertices, blocks and other different parameters of G.
One application of block split subdivision domination is that of prisoners and guards.For security and their over all development of prisoners, each prisoner must be seen by some guards, the concept is that of domination.However in order to develop good attitudes of prisoners, we may require that of subdivision graph, also require each prisoner is taken care by the higher official for overall development, the concept is that of block split subdivision domination.
We need the following theorems for our further results.

Main Results
Here, we will consider several domination parameters and their relation to the block split subdivision domination number.The following results appeared in [6].
Theorem 1: A split dominating set D of G is minimal of each vertex.One of the following conditions holds.i) There exists a vertex ii) is an isolated vertex in iii) is connected.
Theorem 2: For any graph .

Upper bound for
In this section we compare the parameter with the blocks, domination and end blocks of G.

Nordhaus-Gaddum type results
15: For any graph G with and then .

Conclusion
In this paper we found an upper bound for block split subdivision domination in graphs in terms of blocks, domination, end blocks, cut vertices, minimum number of vertices, edge covering of G and maximum independence number of G. Further the sharp lower bounds were found interms of cototal domination, total domination and maximum independence (vertex/edge) number of G. Finally Nordhaus-Gaddum type results are also found.

Theorem 3 : 4 :Case 1 :Case 2 :Theorem 5 :Theorem 8 :
For any connected graph with and , then Proof: Suppose is a complete graph.Then by definition, split domination does not exists.Hence Let be the number of blocks in G and be the blocks in We consider the following cases.Case 1: Suppose there exists atleast one as an edge.Let be a set of vertices corresponding to the blocks of In , generates and as blocks in S(G) such that .Clearly .Consider and .If every vertex of is adjacent to at least one vertex in and if there exist more than one component in , then is a split dominating set.Further if where gives one component in .Then is a minimal split dominating set of Hence which gives Case 2: Suppose such that each is an edge.Then where {q} is the number of edges in As in case 1, which gives Next we discuss in case 3, about the blocks of G such that each block contains at least three vertices.Case 3: Suppose such that each is not an edge.Let be the set of vertices corresponding to the blocks of Consider and .If every vertex of adjacent to at least one vertex in and has more than one component, then gives the split domination number by the definition of block graph which is always less than the number of blocks in G. Hence Theorem For any graph (p,q) with N-end blocks and then Proof: In view of the definition, we consider only those graphs which are not complete in Let be the vertices in G and every vertex of is adjacent to atleast one vertex of , then To establish the upper bound for in terms of and the number of end blocks in G.We consider the following cases.Suppose each block of G is an edge.Then in the number the blocks is 2q.least one vertex of D gives a disconnected graph.Hence is a .In, the number of blocks is more than that of G. On comparing the cardinality of and the number of end blocks N, we have which gives Suppose at least one block of G is not an edge.Then there exists a block with atleast three vertices, for this block is more than as in case 1. Again remains constant.Hence gives Further we compared our concept to the domination and cut vertices of subdivision of G.For any connected , then Proof: For any connected graph G, by the definition of split domination We have the following cases.Proof: To establish the required results, we have the graphs Let be the vertices of G and be For any graph then where is the minimum edge covering number of G. Proof: By the definition of split domination, Let the blocks of G and be the blocks in Let be the set of vertices in which corresponds to the blocks of Let and be the set of vertices which are incident with the edges of B and if then B itself is an edge covering number.Otherwise, consider the minimum number of edges such that forms a minimal edge covering set of are cut vertices in Again Let be the end vertices in Then every vertex in is adjacent to atleast one vertex in H and has more than one component.Then is a split dominating set, distance at most two, covers all the edges in G. Clearly Further, if for any vertex itself is an independent vertex set, otherwise , where and forms a maximum independent set of G with Let , where and be the minimal set of vertices which covers all the vertices in G. Clearly forms a minimal of G. Suppose the subgraph has more than one component, then attach the minimum number of vertices where which are between the vertices of S such that forms exactly one component in the sub graph ,clearly forms a minimal Let be the set of vertices in If where such that and is a independent set in G which gives , which gives = and Clearly which gives the above result.Theorem 14: For any connected graph G, then where is the maximum edge independence number of G. Proof: By the definition of split domination, we considered the graphs with the that Let be the maximal set of edges with for every and Clearly E forms a maximal independent edge set in G. Then Let be the blocks in G and be the blocks in Let be the set of vertices which corresponds to the blocks of .Let D be a dominating set of then there exist a disconnected graph.Hence D is minimal split dominating set in .Hence Now for a vertex with then which gives .