Weakly Convex and Weakly Connected Independent Dominations in the Corona of Graphs

In this paper we constructed a connected graph with a preassigned order, weakly convex domination number, convex domination number, weakly connected independent domination number, and upper weakly connected independent domination number; characterized the weakly convex set, the weakly convex dominating set, and the weakly connected independent dominating set of the corona graph. As direct consequence, the weakly convexity number, weakly convex domination number, and the weakly connected independent domination number of this graph were obtained. Mathematics Subject Classification: 05C12 1516 Rene E. Leonida and Sergio R. Canoy, Jr.


Introduction and Preliminary Results
Let is called weakly convex in G if for every two vertices u, v ∈ C, there exists a u-v geodesic whose vertices belong to C, or equivalently, if for every two vertices u, v ∈ C, d C (u, v) = d G (u, v), where C is the graph induced by C. A set C ⊆ V (G) is called convex in G if for every two vertices u, v ∈ C, the vertex-set of every u-v geodesic is contained in C. The weak convexity number of G, denoted by wcon(G), is the cardinality of a maximum weakly convex proper subset of V (G). A subset S of V (G) is an independent set if for every x, y ∈ S, xy / ∈ E(G). A subset S of V (G) is called weakly connected if the subgraph S w = (N G [S], E W ) weakly induced by S, is connected, where E W is the set of all edges with at least one vertex in S.
A subset S of V (G) is a dominating set of G if for every v ∈ V (G)\S, there exists u ∈ S such that uv ∈ E(G). The domination number γ(G) of G is the smallest cardinality of a dominating set of G. A dominating set of G which is also weakly convex (respectively, convex) is called a weakly convex (respectively, convex ) dominating set of G. The weakly convex (respectively, convex ) domination number γ wcon (G) (respectively, γ con (G)) of G is the smallest cardinality of a weakly convex (respectively, convex) dominating set of G. Lemanska [5] discussed the relationship of the convex and weakly convex domination numbers, including their relationships with other domination parameters. In [4], Janakiraman and Alphonse found bounds for the weakly convex domination number of a graph and its complement in terms of degree, diameter and other graph parameters, and characterized graphs for which bounds are attained.
A dominating set of G which is also independent is called is an independent dominating set of G. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. An independent dominating set of G which is weakly connected is called a weakly connected independent dominating set of G.
The weakly connected independent domination number i w (G) of G is the smallest cardinality of a weakly connected independent dominating set of G. Similarly, the upper weakly connected independent domination number β w (G) is the largest cardinality of a weakly connected independent dominating set of G. In [1], Dunbar, et al., showed that every connected graph has a weakly connected independent dominating set. Thus, for a connected graph G, the weakly connected independent domination number i w (G) and the upper weakly connected independent domination number β w (G) always exist. Relations of these parameters with other domination parameters are also given.
Let G and H be graphs of order m and n, respectively. The corona G • H of G and H is the graph obtained by taking one copy of G and m copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. For every v ∈ V (G), denote by H v the copy of H whose vertices are attached one by one to the vertex v. Denote by v + H v the subgraph of the corona G • H corresponding to the join {v} + H v .
In this paper, we assume that G = (V (G), E(G)) is a simple undirected graph. A path with vertices v 1 , v 2 , ..., v n , with endpoints v 1 and v n is denoted by The following results can be easily verified.
Proposition 1.2 Let n ≥ 2 be a positive integer. Then i w (P n ) = n 2 and β w (P n ) = n 2 .
We construct a graph with a preassigned order and some domination parameters. Theorem 1.3 (Realization Problem) Given positive integers a, b, c, d, and n with 4 ≤ a < b < c < d < n, there exists a graph G with |V (G)| = n, γ wcon (G) = a, γ con (G) = b, i w (G) = c, and β w (G) = d.

Corollary 1.4
The parameter pairs i w and γ wcon , i w and γ con , β w and γ wcon , and β w and γ con are not comparable.

Weakly Convexity
The next result characterizes the weakly convex sets of G • H.
Proof : Suppose that C is a weakly convex set of G • H. Consider the following cases: Since C is weakly convex, there exists a-b geodesic P (a, b) such that V (P (a, b)) ⊆ C. This implies that v, w ∈ C, which contadicts the assumption. Thus, C ⊆ V (H v ) for some v ∈ V (G). If |C| = 1, then diam H v ( C ) ≤ 2. Suppose |C| ≥ 2 and let x, y ∈ C with x = y. Since C is weakly convex, there exists x-y geodesic P (x, y) such that V (P (x, y)) ⊆ C. It follows that diam For the converse, suppose that C ∩ V (G) is a weakly convex set of G, As a consequence of Theorem 2.1, we have

Weakly Convex Domination
The following result characterizes the weakly convex dominating sets of G • H.
Proof : Let C be a weakly convex dominating set of G • H. Then C ∩ V (G) = ∅. Hence, by Theorem 2.1(a), This implies that C does not dominate V (H v ), contrary to our assumption. Therefore, For the converse, suppose that is a weakly convex dominating set of G, it follows from Theorem 2.1 that C is a weakly convex dominating set of G • H.
The next result follows from Theorem 3.1.

Weakly Connected Independent Domination
The next result characterizes the weakly connected independent dominating sets of G • H.
Proof : Suppose that C is a weakly connected independent dominating set of G • H. It is easy to see that C ∩ V (G) is weakly connected and an independent set of G, and C ∩ V (v + H v ) is an independent dominating set of v + H v for every v ∈ V (G). Suppose C ∩ V (G) is not a dominating set of G. Then there exist u ∈ V (G)\C such that uz / ∈ E(G) for all z ∈ C ∩ V (G). This means that C ∩ V (u + H u ) w is a component of C w . This implies that C w is not connected, contrary to the assumption. Thus, C ∩ V (G) is a dominating set of G.
Conversely, suppose that C ∩ V (G) is a weakly connected independent dominating set of G and C ∩ V (v + H v ) is an independent dominating set of v + H v for every v ∈ V (G). Then C is a weakly connected and dominating set of G • H. Let x, y ∈ C with x = y. If x, y ∈ C ∩ V (G), then xy / ∈ E(G • H).
. This shows that C is an independent set of G • H. Proof : Suppose that C is a minimum weakly connected independent dominating set of G • H. Let C 1 = C ∩ V (G) and suppose it is not a maximum weakly connected independent dominating set of G. Let M 1 be a maximum weakly connected independent dominating set of G. Then n = |M 1 | − |C 1 | > 0. For This contrdicts the hypothesis that C is a minimum weakly connected independent dominating set of G • H. Therefore, C ∩ V (G) is a maximum weakly connected independent dominating set of G. Proof : The corollary clearly holds when i(H) = 1. Suppose i(H) = 1. Let C be a minimum weakly connnected independent dominating set of G • H. Let C 1 = C ∩ V (G) and C 2 = C\C 1 . For each u ∈ V (G)\C 1 , let D u ⊆ V (H u ) be an independent dominating set of H u . Then C 2 = {D u : u ∈ V (G)\C 1 }. By Theorem 4.2, C 1 is a maximum weakly connected independent dominating set of G. Thus, |C 1 | = β w (G). Hence, i w (G • H) = |C| = |C 1 | + u∈V (G)\C 1 |D u | ≥ β w (G) + (m − β w (G))i(H).
Next, let C 1 be a maximum weakly connected independent dominating set of G and D be a minimum independent dominating set of H. For each v ∈ V (G)\C 1 , let D v ⊆ V (H v ) be such that D v ∼ = D . Let C 2 = {D v : v ∈ V (G)\C 1 }. By Theorem 4.1, C = C 1 ∪ C 2 is a weakly connected independent dominating set of G • H. Thus, Therefore, i w (G • H) = β w (G) + (m − β w (G))i(H).