Proposition 2

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Conversely, if dinj(J(G)) = 1. And suppose J(G) has no Inj-isolated vertex,then by Theorem 2.21 inj(J(G)) , so if we suppose D is a minimal Inj-dominating set I J(G), then V – D is also a minimal Inj-dominating set. Thus dinj(J(G)) 2, a contradiction. Therefore J(G) has at last one Inj-isolated vertex. iii) Since for every vertex v of the wheel graph the degionj{v} = p –1. Hence dinj(J(Wp)) = p (iv) and (v) the proof is obvious. Evidently each CN-dominating set in J(G) is an Inj-dominating set in J(G) and any CN-domatic partitionis an Inj-domatic partition. Wehave the following proposition.

Proposition 3.3.: For any graph J(G) , dinj(J(G)) dcn(J(G)).

Theorem 3.4 : For any graph J(G) with p vertices , dinj(J(G)) ⁄ inj(J(G))

Proof: Assume that dinj(J(G)) = d and { D1, D2, D3…….Dd} is a partition of V(J(G)) into

d numbers of Inj-dominating sets, clearly | Di | inj(J(G)) for I = 1,2,……d. n we have p= ∑ I | d inj(J(G)), Hence dinj(J(G)) ⁄ inj(J(G))

Theorem 3.5: For any graph J(G) with p vertices, dinj(J(G)) ⁄ δinj(J(G)) ┘ Proof: Let D be any subset of V(J(G)) such that | D | p –δinj(J(G)). For any vertex v V –D we have | Ninj[v} | 1 + δinj(J(G)). Therefore Ninj(v) D ɸ. Thus D is an Inj-dominating set of J(G). So we can take any | ⁄ δinj(J(G)) ] disjoint subset ach of cardinality p –δinj(J(G)). Hence

dinj(J(G)) ⁄ δinj(J(G)) ┘ Theorem 3.6: For any graph such that dinj(J(G)) δinj(J(G)) + 1. Further the equality holds If J(G) is complete graph J(Kp))

Proof: Let J(G) be a graph such that dinj(J(G)) > δinj(J(G)) + 1. Then there exists at least δinj(J(G)) + 2 Inj-dominating sets which they are mutually6 disjoint. Let v be any vertex in V(J(G)) such that deginj(J(G)) = δinj(J(G)). Then there is at least oneof the Inj-dominating sets which has no intersection with NIONJ[v]. Hence, that Inj-dominating set can not dominate v, a contradiction. Therefore dinj(J(G)) δinj(J(G)) + 1. It is a obvious if J(G) is complete, thendinj(J(G)) > δinj(J(G)) + 1.

Theorem 3.7: For any graph J(G) with p vertices dinj(J(G)) + dinj(J( inj) p + 1.

Proof:From Theorem 3.6, we havedinj(J(G)) δinj(J(G)) + 1. and dinj(J( inj)) δinj(J( ̅ inj)) + 1, and clearly δinj(J( ̅ inj)) = p –1 –Δinj(J(G)). Hence

dinj(J(G)) + dinj(J( inj)) δinj(J(G)) + p –Δinj(J(G)) + 1 p + 1

Theorem 3.8: For any graph J(G) with p vertices and without Inj-isolated vertices, dinj(J(G)) + inj(J(G)) p + 1.

Proof: Let J(G) be a graph with p vertices. Then by Theorem 2.22, we have

inj(J(G)) ≤ p –Δinj(J(G)) ≤ p –δinj(J(G)),

And also from Theorem 3.6, dinj(J(G)) ≤ inj (J(G))+ 1.Then

dinj(J(G)) + inj(J(G)) ≤ inj (J(G))+ 1.+ p –δinj(J(G))

Volume: 08 Issue: 06 | June 2021 www.irjet.net p-ISSN: 2395-0072

Hence,

dinj(J(G)) + inj(J(G)) ≤ p + 1

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