We need the following Theorems for our further results: MAIN RESULTS

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1 International Journal of Technical Research Applications e-issn: , SPLIT BLOCK SUBDIVISION DOMINATION IN GRAPHS MH Muddebihal 1, PShekanna 2, Shabbir Ahmed 3 Department of Mathematics, Gulbarga University, Gulbaarga mhmuddebihal@yahoocoin, 2 shaikshavali71@gmailcom, 3 glbhyb09@rediffmailcom Abstract: A dominating set is a split dominating set in If the induced subgraph is disconnected in is denoted by a split dominating set in The split domination number of, is the minimum cardinality of In this paper, some results on were obtained in terms of vertices, blocks, other different parameters of but not members of Further, we develop its relationship with other different domination parameters of Key words: Block graph, Subdivision block graph, split domination number [I] INTRODUCTION All graphs considered here are simple, finite, nontrivial, undirected connected As usual denote the number of vertices, edges blocks of a graph respectively In this paper, for any undefined term or notation can be found in F Harary [3] G Chartr PingZhang [2] The study of domination in graphs was begin by OOre [5] CBerge [1] As usual, The minimum degree maximum degree of a graph are denoted by respectively A vertex cover of a graph is a set of vertices that covers all the edges of The vertex covering number is a minimum cardinality of a vertex cover in The vertex independence number is the maximum cardinality of an independent set of vertices A edge cover of is a set of edges that covers all the vertices The edge covering number of is minimum cardinality of a edge cover The edge independence number of a graph 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 The Domination number of is the minimum cardinality of a dominating set in A dominating set of a graph is a split dominating set if the induced subgraph is disconnected The split domination number of a graph is the minimum cardinality of a split dominating set This concept was introduced by A dominating set of is a cototal dominating set if the induced subgraph has no isolated vertices The cototal domination number is the minimum cardinality of a cototal dominating set See [4] The following figure illustrate the formation of of a graph The domination of split subdivision block graph is denoted by In this paper, some results on obtained in terms of vertices, blocks other parameters of We need the following Theorems for our further results: [II] MAIN RESULTS Theorem A [4]: A split dominating set of is minimal for each vertex one of the following condition holds There exists a vertex is an isolated vertex in is connected Theorem B [4]: For any graph Now we consider the upper bound on blocks in of in terms of Theorem 21: For any graph with 82 P a g e

2 International Journal of Technical Research Applications e-issn: , Proof: For any graph with, a split domination Case2: each block of is a complete graph does not exists Hence we required blocks with we consider the sub cases of be the number of blocks of case 2 with corresponding to the blocks of Also be the set of vertices in, 1 be a set of cut vertices consider a subset such that,1 are not cut vertices is a dominating set Clearly is disconnected graph Then is a Hence Subcase21: Assume is an isolates Hence Then Sub case 22: Assume every block of ) is = In the following Theorem, we obtain an upper bound for in terms of vertices added to Theorem 22: For any connected graph with blocks, is the number of vertices added to Proof: For any nontrivial connected graph has If the graph Then by the definition, split domination set does not exists Hence corresponds to the blocks of following cases Now we consider the Case1: each block of is an edge Then be the set of vertices of Now consider, is a set of cut vertices in are adjacent to end vertices of there exists a subset of with the property is adjacent to atleast one vertex of is a disconnected graph Hence is a By Theorem 1, is an isolate Hence We establish an upper bound involving the Maximum degree the vertices of for split block sub division domination in graphs Theorem 23: For any graph Proof: For split domination, We consider the graphs with the property corresponding to the blocks of be a there exist a vertex By Theorem A, each vertex is a split dominating set in Thus Since by Theorem B The following lower bound relationship is between split domination in vertex covering number in 83 P a g e

3 International Journal of Technical Research Applications e-issn: , Theorem 24: For any graph with, A relationship between the split domination in, is a vertex covering independence number of a graph is established in the number of following theorem Proof: We consider only those graphs are not correspondes to the set B(G), Hence is disconnected, Now each edge in Clearly is adjacent to atleast one vertex in Hence The following result gives a upper bound for of domination end blocks in Theorem 25: For any connected graph with Proof: graph in terms is a block Then by definition, the split domination does not exists Now assume with at least two blocks be the set of blocks in is a graph corresponds to the blocks of Now be the vertices in D is a of, whose vertex set is Note that at least one More over, any component of is of size atleast two Thus is minimal Every vertex of there exists a vertex is adjacent to at least one vertex of every vertex of is not adjacent to at least one vertex Thus Hence Theorem26: For any connected graph with is the independence number of Proof: By the definition of split domination, corresponds to the vertices of the set in be the set of vertices in e have the following cases Case1: is a tree are cut vertices in,were Then we consider with the property is a set of all end vertices in every is an isolatesthus Case 2: cases of case 2 is not a tree we consider sub Subcases21: Assume is a block Then in Thus Hence the number of isolates in, One can see that for the as in case, We have Sub case 22: Assume has atleast two blocksthen as in subcase 21,we have The next result gives a lower bound on the diameter of in terms of Theorem 27: For any graph with blocks, 84 P a g e

4 International Journal of Technical Research Applications e-issn: , Proof : be the Further blocks of,then be the corresponding block vertices in B(G) be the set of edges constitutes the diameteral path in are non end blocks in cut vertices in vertices in are cut Since they are non end blocks in Then is a Clearly Next, the following upper bound for split domination in is interms of edge covering number of Theorem29: For any connected ( graph with is the edge covering number Proof: For any non trivial connected graph with by definition of split domination, the split domination set does not exists Hence is cyclic there exists atleast one block contains a block diametrical path of length atleast two In the block as a singleton if atmost two elements of diameter of is acyclic each edge of is a block of Now, gives Clearly we have The following result is a relationship between, domination vertices of Theorem 28: For any graph with Proof: the graph domination does not exists Hence has one block, split Then be the corresponding block vertices in Also be the set of vertices in is adjacent to atleast one vertex of the set of vertices in Now Then we have Since each element of be are non end blocks in corresponds to the elements of forms a minimal dominating set of is a cut vertex, correspondes to the set We have the following cases Case 1: each block is an edge in Then is the set of end edges, If every cut vertex of is adjacent with an end vertex Then Then atleast one cut vertices in Otherwise there exist are non cut vertices in The is a split dominating set Hence Since has more than one component Hence Case2: has atleast one block is not an edge, be the set of cut vertices vertices in, be the set of cut, Hence is disconnected, As in case 1, will increase Hence 85 P a g e

5 International Journal of Technical Research Applications e-issn: , The following lower bound for split domination in is are cut vertices in interms of edge independence number in is adjacent to at least one vertex in Theorem 210: For any graph with Then gives disconnected graph Thus Proof: By the definition of Split domination, we need We have the following cases Case 1: each block in is an edge Also Consider Then be the set of edges in be a set of alternative edges in, again be the cut vertices are adjacent to at least one vertex of are the end vertices in is disconnected Then Case2: there exists at least one block is not an edge be the set of edges in edges in is the set of alternative be the vertices of Then is a set of cut vertices is a set of non cut vertices Now we consider such that has more than one component Hence is a In the following theorem, we expressed the lower bound for in terms of cut vertices of Theorem 211: For any connected graph with vertices in is the cut Proof: graph is a block Then by the definition, of split domination, consider the following cases Case 1: each block of consider is an edge Then we be the cut vertices in Case 2: each block in Clearly is not an edge be the cut vertices in are the non cut vertices in Further we consider Finally, the following result gives an lower bound on in terms of is disconnected Theorem 212: For any nontrivial tree with Proof: We consider only those H graphs are not, be a subset of are end vertices in with has no isolates, consider be the set of all vertices of with the property, is a set of all end vertices in gives minimum split domination in Clearly REFERENCES [I] C Berge, Theory of graphs its applications, Methuen, London, (1962) [II] G Chartr Ping Zhang, Introduction to graph Theory, Newyork (2006) [III] FHarary, Graph Theory, Adison Wesley, Reading Mass (1972) [IV] VRKulli, Theory of domination in Graphs, Vishwa international Publications, Gulbarga, India (2010) [V] OOre, Theory of graphs, Amer Math soc, Colloq Publ, 38 Providence, (1962) 86 P a g e