Vol 8, No 2 (2017)

Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 8(2), Feb. – 2017 184

ON THE ROMAN DOMINATION NUMBER OF GRAPHS

D. K. THAKKAR

1

Department of Mathematics,

Saurashtra University Campus, University Road, Rajkot – 360005, India.

S. M. BADIYANI*

2

Department of Mathematics,

Saurashtra University Campus, University Road, Rajkot – 360005, India.

(Received On: 11-01-17; Revised & Accepted On: 28-02-17)

ABSTRACT

I

n this manuscript we consider the change in the Roman Domination Number of a graph when a vertex is removed from the graph. We prove a necessary and sufficient condition under which the Roman Domination Number of a graph increases. Also we prove a necessary and sufficient condition under which the Roman Domination Number decreases. We deduce that in any graph there are vertices whose removal does not increase the Roman Domination Number.

Keywords: Roman Dominating Function, Roman Domination Number, Minimal Roman Dominating Function,

Minimum Roman Dominating Function, Private Neighbourhood.

AMS Mathematics Subject Classification (2010): 05C69.

1. INTRODUCTION

The concept of Roman Domination was introduced in [5] by Ernie J. Cockayne, T.W.Haynes, and others. A Roman Dominating Function gives rise to a Dominating Function and vice-versa. In [5] a necessary and sufficient condition has been proved under which the Domination Number of a graph increases when a vertex is removed from the graph. A necessary and sufficient condition was also proved under which the Domination Number decreases when a vertex is removed from the graph [5]. Here we prove the conditions for the same operations for the Roman Domination Number.

2. PRELIMINARIES AND NOTATIONS

In this paper we consider only those graphs which are simple and finite. If𝐺𝐺is a graph, 𝑉𝑉(𝐺𝐺) will denote the vertex set of graph𝐺𝐺 and 𝐸𝐸(𝐺𝐺) will denote the edge set of graph𝐺𝐺. If𝐺𝐺is a graph and 𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) then𝐺𝐺 − 𝑣𝑣 will denote the subgraph obtained by removing the vertex𝑣𝑣from𝐺𝐺. The Roman Domination Number of the graph𝐺𝐺 is denoted as 𝛾𝛾_𝑅𝑅(𝐺𝐺), whereas the Domination Number of the graph G is denoted as𝛾𝛾(𝐺𝐺). If 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{0,1,2} is a function then we write,

𝑽𝑽𝟐𝟐(𝒇𝒇) = {𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) / 𝑓𝑓(𝑣𝑣) = 2}

𝑽𝑽𝟏𝟏(𝒇𝒇) = {𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) / 𝑓𝑓(𝑣𝑣) = 1}

𝑽𝑽𝟎𝟎(𝒇𝒇) = {𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) / 𝑓𝑓(𝑣𝑣) = 0}

Obviously the above sets are mutually disjoint and their union is the vertex set V(G). The weight of this function

𝑓𝑓=∑𝑣𝑣∈𝑉𝑉(𝐺𝐺)𝑓𝑓(𝑣𝑣). This number is denoted as𝑤𝑤(𝑓𝑓). We will also use the following notations: 𝑽𝑽𝟎𝟎_{= {𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) / 𝛾𝛾}

𝑅𝑅(𝐺𝐺 − 𝑣𝑣) =𝛾𝛾𝑅𝑅(𝐺𝐺)}

𝑽𝑽+_{= {𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) / 𝛾𝛾}

𝑅𝑅(𝐺𝐺 − 𝑣𝑣) >𝛾𝛾𝑅𝑅(𝐺𝐺)}

𝑽𝑽−_{= {𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) / 𝛾𝛾}

𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾𝑅𝑅(𝐺𝐺)}

If 𝑆𝑆 ⊂ 𝑉𝑉(𝐺𝐺) and𝑣𝑣 ∈ 𝑆𝑆, then the private neighbourhood of 𝑣𝑣 with respect to the set S = {𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) / 𝑁𝑁[𝑢𝑢]∩ 𝑆𝑆= {𝑣𝑣}}. It is denoted as𝑝𝑝𝑣𝑣[𝑣𝑣,𝑆𝑆].

Corresponding Author: S. M. Badiyani*

2

_{, Department of Mathematics,}

Definition 2.1 [4]: Let 𝐺𝐺 be a graph. A function 𝑓𝑓: 𝑉𝑉(𝐺𝐺)→{0, 1, 2} is called a Roman Dominating Function if every vertex 𝑢𝑢 for which 𝑓𝑓(𝑢𝑢) = 0 is adjacent to at least one vertex 𝑣𝑣 for which𝑓𝑓(𝑣𝑣) = 2.

Definition 2.2 [4]: A Roman Dominating Function with minimum weight is called a Minimum Roman Dominating

Function.

Definition 2.3 [4]: The weight of a Minimum Roman Dominating Function is called the Roman Domination Number of

the graph. It is denoted as 𝛾𝛾_𝑅𝑅(𝐺𝐺).

Definition 2.4 [4]: Let 𝐺𝐺 be a graph. A function 𝑓𝑓: 𝑉𝑉(𝐺𝐺)→{0, 1, 2} is called a Minimal Roman Dominating Function

if (𝑖𝑖) 𝑓𝑓 is a Roman Dominating Function. (𝑖𝑖𝑖𝑖) Whenever 𝑔𝑔: 𝑉𝑉(𝐺𝐺)→{0, 1, 2} and 𝑔𝑔<𝑓𝑓 then 𝑔𝑔 is not a Roman Dominating Function.

Remark 2.5:

i) For any graph𝐺𝐺,𝛾𝛾(𝐺𝐺)≤ 𝛾𝛾_𝑅𝑅(𝐺𝐺)≤2𝛾𝛾(𝐺𝐺).

ii) In [2] it is proved that a Roman Dominating Function𝑓𝑓 is minimal if and only if each of the following two conditions is satisfied:

1)

If 𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) and 𝑓𝑓(𝑣𝑣) = 2 then there is a vertex 𝑥𝑥 such that𝑓𝑓(𝑥𝑥) = 0, 𝑥𝑥 is adjacent to 𝑣𝑣 and 𝑥𝑥 is not adjacent to any other vertex 𝑤𝑤 for which𝑓𝑓(𝑤𝑤) = 2.

2)

If 𝑓𝑓(𝑤𝑤) = 1 then 𝑤𝑤 is not adjacent to any vertex 𝑥𝑥 for which𝑓𝑓(𝑥𝑥) = 2.

3. VERTEX REMOVAL AND ROMAN DOMINATION NUMBER

Proposition 3.1: Let𝐺𝐺 be a graph and𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺). If 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺), then 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) =𝛾𝛾_𝑅𝑅(𝐺𝐺)−1.

Proof: Let𝑔𝑔 be a Minimum Roman Dominating Function of𝐺𝐺 − 𝑣𝑣, then 𝑤𝑤(𝑔𝑔) <𝛾𝛾_𝑅𝑅(𝐺𝐺).

Now define 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{0,1,2} as follows:

𝑓𝑓(𝑣𝑣) = 1 and

𝑓𝑓(𝑤𝑤) =𝑔𝑔(𝑤𝑤);𝑖𝑖𝑓𝑓𝑤𝑤 ≠ 𝑣𝑣

Obviously𝑓𝑓 is a Roman Dominating Function on𝐺𝐺 and𝑤𝑤(𝑓𝑓) =𝑤𝑤(𝑔𝑔) + 1.

Since 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺),𝑓𝑓 must be a Minimum Roman Dominating Function of𝐺𝐺.

Thus 𝛾𝛾𝑅𝑅(𝐺𝐺) =𝑤𝑤(𝑓𝑓) =𝑤𝑤(𝑔𝑔) + 1 = 𝛾𝛾𝑅𝑅(𝐺𝐺 − 𝑣𝑣) + 1.

i. e. 𝛾𝛾𝑅𝑅(𝐺𝐺 − 𝑣𝑣) = 𝛾𝛾𝑅𝑅(𝐺𝐺)−1.

Theorem 3.2: Let𝐺𝐺 be a graph and𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) then 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺) if and only if there is a Minimum Roman Dominating Function𝑓𝑓 on𝐺𝐺 such that𝑓𝑓(𝑣𝑣) = 1.

Proof: First suppose 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺). Let𝑔𝑔 be a Minimum Roman Dominating Function of𝐺𝐺 − 𝑣𝑣. Now define

𝑓𝑓:𝑉𝑉(𝐺𝐺)→{0,1,2} as follows:

𝑓𝑓(𝑣𝑣) = 1 and

𝑓𝑓(𝑤𝑤) =𝑔𝑔(𝑤𝑤);𝑖𝑖𝑓𝑓𝑤𝑤 ≠ 𝑣𝑣

Obviously𝑓𝑓 is a Minimum Roman Dominating Function on𝐺𝐺 and𝑓𝑓(𝑣𝑣) = 1.

Conversely suppose there is a Minimum Roman Dominating Function𝑓𝑓 such that𝑓𝑓(𝑣𝑣) = 1.

Define 𝑔𝑔:𝑉𝑉(𝐺𝐺 − 𝑣𝑣)→{0,1,2} as follows:

𝑔𝑔(𝑤𝑤) =𝑓𝑓(𝑤𝑤);∀𝑤𝑤 ∈ 𝑉𝑉(𝐺𝐺 − 𝑣𝑣)

Then obviously𝑔𝑔 is a Roman Dominating Function on𝐺𝐺 − 𝑣𝑣 and 𝑤𝑤(𝑔𝑔) <𝑤𝑤(𝑓𝑓) =𝛾𝛾_𝑅𝑅(𝐺𝐺).

Therefore 𝛾𝛾𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾𝑅𝑅(𝐺𝐺).

Corollary 3.3: Let 𝐺𝐺 be a graph and 𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) be an isolated vertex then 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺).

Proof: For any Minimum Roman Dominating Function𝑓𝑓 on𝐺𝐺,𝑓𝑓(𝑣𝑣) = 1 as 𝑣𝑣 is an isolated vertex [2].

© 2017, IJMA. All Rights Reserved 186 Consider the following example in which the vertex set is {v₁, v₂, v₃, v₄}.

Figure-1: (GRAPH G)

Let 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{0,1,2} be any function such that

𝑓𝑓(𝑣𝑣1) = 0,𝑓𝑓(𝑣𝑣2) = 0,𝑓𝑓(𝑣𝑣3) = 2 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓(𝑣𝑣4) = 0

The Roman Domination Number of the graph𝐺𝐺is 2;

Whereas the Roman Domination Number of 𝐺𝐺 − 𝑣𝑣₃ is 3.

Thus we have 𝜸𝜸_𝑹𝑹(𝑮𝑮 − 𝒗𝒗_𝟑𝟑) >𝜸𝜸_𝑹𝑹(𝑮𝑮).

Now we state and prove the necessary and sufficient conditions under which the Roman Domination Number increases when a vertex is removed from the graph.

Theorem 3.4: Let𝐺𝐺 be a graph and𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) then 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) >𝛾𝛾_𝑅𝑅(𝐺𝐺) if and only if the following conditions are satisfied:

i) 𝑣𝑣 is not an isolated vertex in𝐺𝐺.

ii) 𝑓𝑓(𝑣𝑣) = 2 for every Minimum Roman Dominating Function𝑓𝑓 on𝐺𝐺.

iii) There is no Roman Dominating Function𝑔𝑔 on 𝐺𝐺 − 𝑣𝑣such that𝑤𝑤(𝑔𝑔)≤ 𝛾𝛾_𝑅𝑅(𝐺𝐺) and 𝑉𝑉₂(𝑔𝑔) is a subset of

𝑉𝑉(𝐺𝐺)− 𝑁𝑁[𝑣𝑣].

Proof: Suppose 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) >𝛾𝛾_𝑅𝑅(𝐺𝐺)

i) Suppose𝑣𝑣 is an isolated vertex, then for every minimum function𝑓𝑓on 𝐺𝐺,𝑓𝑓(𝑣𝑣) = 1 and therefore 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <

𝛾𝛾𝑅𝑅(𝐺𝐺) by Theorem 3.2; which is a contradiction.

Therefore𝑣𝑣 is not isolated vertex in𝐺𝐺.

ii) Suppose for some minimum function𝑓𝑓on 𝐺𝐺,𝑓𝑓(𝑣𝑣) = 0. Now define𝑔𝑔 on𝐺𝐺 − 𝑣𝑣 as follows:

𝑔𝑔(𝑤𝑤) =𝑓𝑓(𝑤𝑤);∀𝑤𝑤 ∈ 𝑉𝑉(𝐺𝐺 − 𝑣𝑣)

Obviously𝑔𝑔 is a Roman Dominating Function on𝐺𝐺 − 𝑣𝑣.

Then 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣)≤ 𝑤𝑤(𝑔𝑔)≤ 𝑤𝑤(𝑓𝑓) =𝛾𝛾_𝑅𝑅(𝐺𝐺); which is a contradiction.

Suppose for some minimum function𝑓𝑓 on𝐺𝐺,𝑓𝑓(𝑣𝑣) = 1, then 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺) by theorem 3.2; which is a contradiction.

Hence𝑓𝑓(𝑣𝑣) = 2 for every Minimum Roman Dominating Function𝑓𝑓 on𝐺𝐺

iii) Suppose there is a Roman Dominating Function𝑔𝑔 on𝐺𝐺 − 𝑣𝑣 with 𝑤𝑤(𝑔𝑔)≤ 𝛾𝛾_𝑅𝑅(𝐺𝐺) and 𝑉𝑉₂(𝑔𝑔) is a subset of𝑉𝑉(𝐺𝐺)− 𝑁𝑁[𝑣𝑣].

Then obviously𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣)≤ 𝑤𝑤(𝑔𝑔)≤ 𝛾𝛾_𝑅𝑅(𝐺𝐺); which is a contradiction.

Conversely suppose conditions (i), (ii) and (iii) are satisfied. Suppose 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) =𝛾𝛾_𝑅𝑅(𝐺𝐺).

Let𝑔𝑔 be a Minimum Roman Dominating Function on𝐺𝐺 − 𝑣𝑣. Consider the set𝑉𝑉₂(𝑔𝑔). First suppose𝑣𝑣 is adjacent to some vertex of𝑉𝑉₂(𝑔𝑔). Now define 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{0,1,2} as follows:

𝑓𝑓(𝑣𝑣) = 0 and 𝑓𝑓(𝑤𝑤) =𝑔𝑔(𝑤𝑤);∀𝑤𝑤 ≠ 𝑣𝑣

Then𝑓𝑓is a Minimum Roman Dominating Function on𝐺𝐺 and𝑤𝑤(𝑓𝑓) =𝑤𝑤(𝑔𝑔).

But𝑓𝑓(𝑣𝑣) = 0 which contradicts condition (ii).

Thus 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) =𝛾𝛾_𝑅𝑅(𝐺𝐺) is not possible.

Suppose 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺). Let𝑔𝑔 be a Minimum Roman Dominating Function on𝐺𝐺 − 𝑣𝑣.

Letℎ be defined as follows:

ℎ(𝑣𝑣) = 0 and ℎ(𝑤𝑤) =𝑔𝑔(𝑤𝑤);∀𝑤𝑤 ≠ 𝑣𝑣

Then𝑤𝑤(ℎ) =𝑤𝑤(𝑔𝑔). Thereforeℎ can not be a Roman Dominating Function on𝐺𝐺. Therefore there is a vertex𝑥𝑥of𝐺𝐺such thatℎ(𝑥𝑥) = 0, but 𝑥𝑥is not adjacent to any vertex 𝑦𝑦 for whichℎ(𝑦𝑦) = 2. From the definition ofℎ it is clear that𝑥𝑥=𝑣𝑣 and𝑣𝑣 is not adjacent to any vertex𝑤𝑤 for which 𝑔𝑔(𝑤𝑤) = 2.

Now 𝑉𝑉₂(𝑔𝑔)⊂ 𝑉𝑉(𝐺𝐺)− 𝑁𝑁[𝑣𝑣] and also 𝑤𝑤(𝑔𝑔)≤ 𝛾𝛾_𝑅𝑅(𝐺𝐺) and 𝑔𝑔 is a Roman Dominating Function on𝐺𝐺 − 𝑣𝑣; this again contradicts condition (iii).

Therefore 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺) is also not possible.

Thus 𝛾𝛾𝑅𝑅(𝐺𝐺 − 𝑣𝑣) >𝛾𝛾𝑅𝑅(𝐺𝐺) .

Remark 3.5: We may note that if𝑓𝑓 is a Minimal Roman Dominating Function and𝑓𝑓(𝑢𝑢) = 1 and𝑓𝑓(𝑣𝑣) = 2, then𝑢𝑢 and𝑣𝑣 can not be adjacent vertices [2].

Corollary 3.6: Let𝐺𝐺 be a graph and𝑢𝑢,𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) such that 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑢𝑢) >𝛾𝛾_𝑅𝑅(𝐺𝐺) and𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺) then𝑢𝑢 and𝑣𝑣 are non adjacent vertices.

Proof: Since 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) <𝛾𝛾_𝑅𝑅(𝐺𝐺) there is a Minimum Roman Dominating Function𝑓𝑓 such that𝑓𝑓(𝑣𝑣) = 1. Also by theorem 3.4 we have𝑓𝑓(𝑢𝑢) = 2.

So by the above remark 2.5(2) 𝑢𝑢 and𝑣𝑣 cannot be adjacent vertices.

Theorem 3.7: Let𝐺𝐺 be a graph and𝑣𝑣 ∈ 𝑉𝑉(𝐺𝐺) then 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) >𝛾𝛾_𝑅𝑅(𝐺𝐺). Let𝑓𝑓 be a Minimum Roman Dominating Function of𝐺𝐺, then𝑓𝑓(𝑣𝑣) = 2 and there is atleast two distinct vertices 𝑢𝑢₁ and 𝑢𝑢₂ such that𝑓𝑓(𝑢𝑢₁) =𝑓𝑓(𝑢𝑢₂) = 0,

𝑢𝑢1,𝑢𝑢2∈ 𝑝𝑝𝑣𝑣[𝑣𝑣,𝑉𝑉2(𝑓𝑓)] and also 𝑢𝑢1 ,𝑢𝑢2 are non adjacent vertices.

Proof: Since 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑣𝑣) >𝛾𝛾_𝑅𝑅(𝐺𝐺),𝑓𝑓(𝑣𝑣) = 2.Since𝑓𝑓 is a Minimal Roman Dominating Function by theorem 3.1 of [2] there is at least one vertex𝑢𝑢such that𝑓𝑓(𝑢𝑢) = 0 and𝑢𝑢 ∈ 𝑝𝑝𝑣𝑣[𝑣𝑣,𝑉𝑉₂(𝑓𝑓)].Suppose𝑢𝑢 is the only such vertex. Now define𝑔𝑔:𝑉𝑉(𝐺𝐺)→{0,1,2} as follows:

𝑔𝑔(𝑣𝑣) = 0 ,𝑔𝑔(𝑢𝑢) = 2 and

𝑔𝑔(𝑤𝑤) =𝑓𝑓(𝑤𝑤);𝑓𝑓𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑤𝑤 ∈{𝑢𝑢,𝑣𝑣}

If𝑥𝑥 is a vertex different from 𝑢𝑢 and𝑣𝑣, then𝑥𝑥 ∉ 𝑝𝑝𝑣𝑣[𝑣𝑣,𝑉𝑉₂(𝑓𝑓)]. Therefore there is a vertex𝑦𝑦 such that

𝑦𝑦 ≠ 𝑣𝑣,𝑔𝑔(𝑦𝑦) =𝑓𝑓(𝑣𝑣) = 2 and𝑥𝑥 is adjacent to𝑦𝑦. Thus𝑔𝑔 is a Roman Dominating Function. Also𝑤𝑤(𝑔𝑔) =𝑤𝑤(𝑓𝑓). Therefore𝑔𝑔 is a Minimum Roman Dominating Function and𝑔𝑔(𝑣𝑣) = 0; which contradicts theorem 3.4 condition (ii).

Therefore there are at least two vertices say𝑢𝑢 and𝑢𝑢′ such that 𝑓𝑓(𝑢𝑢) =𝑓𝑓(𝑢𝑢′) = 0 and 𝑢𝑢,𝑢𝑢′ ∈ 𝑝𝑝𝑣𝑣[𝑣𝑣,𝑉𝑉₂(𝑓𝑓)].

Suppose any two of them are adjacent. Suppose𝑢𝑢₁and 𝑢𝑢₂are any two vertices such that𝑓𝑓(𝑢𝑢₁) =𝑓𝑓(𝑢𝑢₂) = 0 and

𝑢𝑢1,𝑢𝑢2∈ 𝑝𝑝𝑣𝑣[𝑣𝑣,𝑉𝑉2(𝑓𝑓)].

Now defineℎ:𝑉𝑉(𝐺𝐺)→{0,1,2} as follows:

ℎ(𝑣𝑣) = 0 ,ℎ(𝑢𝑢1) = 2 and

ℎ(𝑤𝑤) =𝑓𝑓(𝑤𝑤);𝑓𝑓𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑓𝑓𝑜𝑜ℎ𝑒𝑒𝑓𝑓𝑣𝑣𝑒𝑒𝑓𝑓𝑜𝑜𝑖𝑖𝑣𝑣𝑒𝑒𝑣𝑣

Let𝑧𝑧 be a vertex such that𝑓𝑓(𝑧𝑧) = 0 and𝑧𝑧 ∈ 𝑝𝑝𝑣𝑣[𝑣𝑣,𝑉𝑉₂(𝑓𝑓)]. If 𝑧𝑧 ≠ 𝑢𝑢₁ then 𝑧𝑧 is adjacent to𝑢𝑢 ₁because of our assumption. Thereforeℎ is a Roman Dominating Function with 𝑤𝑤(ℎ) =𝑤𝑤(𝑓𝑓) and thereforeℎ is a Minimum Roman Dominating Function withℎ(𝑣𝑣) = 0; which is a contradiction.

Thus there must be at least two vertices𝑢𝑢 and𝑢𝑢′ such that 𝑓𝑓(𝑢𝑢) =𝑓𝑓(𝑢𝑢′) = 0 and𝑢𝑢,𝑢𝑢′ ∈ 𝑝𝑝𝑣𝑣[𝑣𝑣,𝑉𝑉₂(𝑓𝑓)] and they are non adjacent.

© 2017, IJMA. All Rights Reserved 188 Since 𝑢𝑢 and𝑣𝑣 are adjacent 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑢𝑢) <𝛾𝛾_𝑅𝑅(𝐺𝐺) is not possible by corollary 3.6. Since𝑓𝑓(𝑢𝑢) = 0, 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑢𝑢) >𝛾𝛾_𝑅𝑅(𝐺𝐺) is not possible by theorem 3.4 condition (ii). Thus it must be true that 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑢𝑢) =𝛾𝛾_𝑅𝑅(𝐺𝐺).

Similarly, 𝛾𝛾_𝑅𝑅(𝐺𝐺 − 𝑢𝑢′) =𝛾𝛾_𝑅𝑅(𝐺𝐺)

Thus every vertex𝑣𝑣 ∈ 𝑉𝑉+ gives rise to at least two vertices in 𝑉𝑉0 and therefore |𝑉𝑉0|≥2|𝑉𝑉+|.

4. REFERENCES

1. C.S.Revelle, K.E.Rosing, Defenders Imperium Romanum: a classical problem in military strategy, Amer. Math. Monthly 107(7) (2000), 585-594.

2. D.K.Thakkar and S.M.Badiyani, “Minimal Roman Dominating Functions” communicated. 3. I. Stewart, Defend the Roman Empire!, Sci. Amer.281 (6) (1999), 136-139.

4. Paul Andrew Dreyer, Jr. Dissertation Director: Fred S Roberts, Application and Variations of domination in graphs, New Brunswick, New Jersey, October (2000).

5. T.W.Haynes, S.T.Hedetniemi, P.J.Slater, Fundamental of Domination In graphs, Marcel Dekker, New York, (1998).

6. T.W.Haynes, S.T.Hedetniemi, P.J.Slater, Domination In graphs Advanced Topics, New York, (1998).

Source of support: Nil, Conflict of interest: None Declared.