Weak Convex Domination in Hypercubes OPEN ACCESS

The n-cube Q n is the graph whose vertex set is the set of all n-dimensional Boolean vectors, two vertices being joined if and only if they differ in exactly one coordinate. The n-star graph S n is a simple graph whose vertex set is the set of all n! permutations of {1, 2, • • • , n} and two vertices α and β are adjacent if and only if α(1)≠β(1) and α(i) ≠β(i) for exactly one i ,i≠ 1. In this paper we determine weak convex domination number for hypercubes. Also convex, weak convex, m - convex and l1-convex numbers of star and hypercube graphs are determined.


Introduction
Graphs considered here are connected, simple. Akers and Krishnamurthy introduced the n-star graph S n [1]. The vertex set of and β are adjacent if and only if α(1) ≠β(1) and α(i) ≠β(i) for exactly one i, i ≠ 1.
The n-star graph is an alternative to n-cube with superior characteristics. Day and Tripathi have compared the topological properties of the n-star and the n-cube in [5]. Arumugam and Kala have determined some domination parameters of star graph and obtained bounds for γ, γ i , γ t , γc and γ p in n-cube for n ≥ 7 in [2].
Let G be a simple connected graph. A subset S of V is called a convex set if for any u, v in S, S contains all the vertices of every u − v geodesic in G. A subset S of V is called a weak convex set if for any u, v in S , S contains all the vertices of a u − v geodesic in G.
A subset S of V is called a m -convex set if for any u, v in S,S contains all the vertices of every u − v induced path in G.
A subset S of V is called al1 -convex set if it is convex and has a vertex which is adjacent to rest of the vertices of S. Maximum

OPEN ACCESS
Volume: 8 cardinality of a proper convex set is the convexity number of G. In a similar way we defi ne weak convex number, m -convex number and l 1 -convex is the maximum of {Con < N[x] > /x εV(G)}.
A subset S of V is called a domination set if every vertex in V − S is adjacent to at least one vertex in S. A dominating set is a weak convex dominating set if it is weak convex. So far exact value of domination number for large n in Q n has not been determined. Here we determine weak convex domination number of Q n for any n.
We know that γ(Q 3 ) = 2. Either {3, 5} or {1, 8} can be chosen that is diametrically opposite vertices are chosen. Therefore their distance is three and hence γ wc (Q 3 )= 4. Let γ wc (Q 3 ) set be {1,  4 . These eight vertices can be chosen in any manner from the two layers of Q 3 . Hence we observe that for Q n , 2 n−1 vertices are required for a weak convex dominating set which can be got in any manner from the two layers of Q n−1 . Now we claim that 2 n−1 is the minimum number of vertices for a weak convex dominating set in Q n .
Let k + l = 2 n−1 where k,lare the number of vertices chosen in two layers of Q n−1 for a weak convex dominating set in Q n .
Without loss of generality assume l < k. Let Q 1 n−1 and Q 2 n−1 denote the fi rst and second layers of Q n−1 . Choose k vertices in Q 1 n−1 in such a way that they form a weak convex dominating set in Q n−1 . . Then single vertex that dominates u and v is either u or v. Therefore weak convexity is violated between u(v) and a vertex among k vertices which is adjacent to private neighbors of u(v) in Q 1 n−1 . Thus a contradiction. If private neighbors of u and v do not form an edge in Q 1 n−1 and N(u)⋂N(v) ≠ φ then weak convexity is violated in Q 2 n−1 which is a contradiction. If private neighbors of u and v do not form an edge in Q 1 n−1 and N(u)⋂N(v) = φ then either u or v is required for domination in Q 2 n−1 . Thus weak convexity is violated between u(v) and a vertex among k vertices which is adjacent to private neighbors of u(v) in Q 1 n−1 . Thus a contradiction. Therefore, k − (l − 1)<2. Hence minimum one vertex must be included in any one of the layers of Q n−1 for a weak convex dominating set in Q n .

Case (ii)
None of l − 1 vertices are private neighbors of k vertices. Clearly weak convexity is violated between any vertex of Q 1 n−1 and Q 2 n−1 .

Case (iii)
Some of l − 1 vertices are private neighbors of k vertices. By Case (i) we get the result. Interchanging k and l we get the result for k < l.

Conclusion
In this paper we determined weak convex domination number for hypercube graphs. We also determined convex, weak convex, m -convex and l 1 -convex numbers of star and hypercube graphs. Other domination parameters for hypercubes are under study in our group.