Computing the split domination number of grid graphs

A setD ⊆ V is a dominating set ofG if every vertex in V −D is adjacent to some vertex inD. The dominating number γ(G) of G is the minimum cardinality of a dominating set D. A dominating set D of a graphG = (V,E) is a split dominating set if the induced graph 〈V −D〉 is disconnected. The split domination number γs(G) is the minimum cardinality of a split domination set. In this paper we have introduced a new method to obtain the split domination number of grid graphs by partitioning the vertex set in terms of star graphs and also we have obtained the exact values of γs(Gm,n),m ≤ n,m, n ≤ 24.


Introduction
The graphs considered here are finite, connected, undirected without loops or multiple edges and without isolated vertices. As usual n and q denote the number of vertices and edges of a graph G. For any undefined term or notation in this paper can be found in Harary [2].
A set D ⊆ V is a dominating set of G if every vertex in V − D is adjacent to some vertex in D. The dominating number γ(G) of G is the minimum cardinality of a dominating set D [5]. V.R.
Kulli and B. Janakiram had introduced a concept of split domination [3]. A dominating set D of a graph G = (V, E) is a split dominating set, if the induced graph V − D is disconnected. The split domination number γ s (G) is the minimum cardinality of a split domination set.
A two dimensional grid graph G m,n is the graph Cartesian product P m × P n of paths on m and n vertices. The Cartesian graph product of G 1 × G 2 with disjoint vertex sets and edge sets in G 1 , G 2 is the graph with the vertex set V 1 × V 2 and two vertices u = (u 1 , u 2 ) and v = (v 1 , v 2 ) are adjacent in G 1 × G 2 whenever [u 1 = v 1 and u 2 adj v 2 ] or [u 2 = v 2 and u 1 adj v 1 ]. A star graph is a complete bipartite graph of the form K 1,n−1 with n vertices. The neighborhood of a vertex in the graph G is the set of vertices adjacent to v and is denoted by N (v).
Computing of domination of grid graph has been studied in [1,4]. In this paper we have introduced a new method to obtain the split domination number of a grid graphs by partitioning the vertex set in terms of K 1,3 , K 1,2 , K 2 and K 1 and also we have obtained the exact values of γ s (G m,n ), m ≤ n, m, n ≤ 24.

Preliminaries
To simplify the description of the algorithm, we first define an order of the vertices of an mn grid graph with vertices v i,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n. Every minimum dominating set can be constructed by an exhaustive search where in each step any undominated vertex is picked, after which all possible ways of dominating this vertex are considered in turn.
The procedure to construct the minimum split dominating set is as follows: such that the number of vertices in D is minimum and < V (G) − D > is disconnected, this procedure is continued unless V (G) − E is an empty set. Suppose if < V (G) − D > is connected, then we need one more vertex to make the graph disconnected.
3. Algorithm to Find the Split Domination Number of grid graph by partitioning the vertex set.
Step 1: Divide the grid graph G m,n by partitioning the vertex set in terms K 1,3 , K 1,2 , K 1,1 and K 1 such that Step 2: Suppose A contains atleast two partition set say P 1 , P 2 such that < V (P 1 ) >= K 1 and Step 3: Step 5: For each partition P j .
Let S be the set of all such vertices.
Step 6: Step 7: Find the split adjacency matrix Otherwise.
Step 8: IF a ij contains atleast one zero row then, γ s = |C| GOTO STEP 11. ELSE GOTO STEP 9.
Step 9: Let {v k } is the row in a ij in which sum of all the elements in {v k } = 1 and 1 is present in v p column.

Examples
Since there exists a zero row v 10 in a ij . therefore the split domination number γ s = |C| = 6. A = {P 1 , P 2 , P 3 , P 4 , P 5 , P 6 , P 7 } Since < V (P 5 ) > and < V (P 6 ) >= K 1 and The Split adjacent matrix a ij is: Since there exists a zero rowv 13 in a ij , γ s = |C| = 6 Figure 3. A grid graph G 4,4 The Split adjacent matrix a ij is: Since there exists a non-zero row in a ij and v 16 contains 1 in v th 15 column=v p , γ s = |C| + 1 = 5 Proof. Let D be the γ-set of G.
Case 1: if V (G) − D is disconnected, the result follows from the definition of split dominating set.

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Computing the split domination number of grid graphs | V.R. Girish and P. Usha