Communications on Applied Nonlinear Analysis 

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Effect of Edge Deletion on the Weak Roman Domination Number of a 

Graph 
 

P. Roushini Leely Pushpam1, M. Kamalam2 

1 Department of Mathematics, 

D. B. Jain College, Chennai - 600097, Tamil Nadu, India. 

e-mail: roushinip@yahoo.com 
2 Shri S. S. Shasun Jain College for Women, 

Chennai - 600017, Tamil Nadu, India. 

e-mail: divinegrace27@gmail.com 

 

 

Article History: 

Received: 26-10-2024 

Revised: 25-11-2024 

Accepted: 22-12-2024 

Abstract:  

Consider a graph Γ with vertex set 𝑉 and edge set 𝐸. For a function 𝑔 defined on 

the vertex set having values in the set {0, 1,2}, the weight of a vertex 𝑥 is 𝑔(𝑥). The weight 

of a subset 𝑋 of 𝑉 is denoted by 𝑔(𝑋) and is defined as the sum of the weights of all the 

vertices in 𝑋. An undefended vertex under 𝑔 is a vertex 𝑎 in Γ with a neighbourhood with 

weight 0.  

A weak Roman Dominating Function 𝑔 (𝑊𝑅𝐷𝐹) is a function such that for any 

vertex 𝑥 having weight 0, there is a vertex 𝑦 in the open neighbourhood of 𝑥 having positive 

weight such that, under the function ℎ defined on 𝑉 having values in the set {0,1,2} defined 

by ℎ(𝑥) = 1, ℎ(𝑦) = 𝑔(𝑦) − 1, ℎ(𝑧) = 𝑔(𝑧), if 𝑧 ∈ 𝑉 − {𝑥, 𝑦}, there are no undefended 

vertices in Γ. The number 𝛾𝑟(Γ), the minimum of the weights of all the 𝑊𝑅𝐷𝐹𝑠 defined 

on Γ is called the weak Roman Domination Number of  Γ. The 𝑊𝑅𝐷𝐹 whose weight is 

𝛾𝑟(Γ) is called a 𝛾𝑟 − function and  𝛾𝑟(Γ) is called the 𝛾𝑟 − value of  Γ. 

With respect to a graph Γ, we say that an edge 𝑥 ∈ 𝐸− if and only if its removal 

will reduce the 𝛾𝑟 − value of Γ. Similarly, we say that an edge 𝑥 ∈ 𝐸+ if and only if its 

removal will increase the 𝛾𝑟 − value of Γ and an edge 𝑥 ∈ 𝐸0 if and only if its removal will 

leave unaltered the 𝛾𝑟 − value of Γ.  

In this paper, we categorize edges of a graph as belonging to 𝐸−, 𝐸+ or 𝐸0.  

Keywords: Weak Roman dominating number, Changing and Unchanging edges, edge 

deletion. 

 

1.  Introduction 

For a graph Γ having vertex set 𝑉 and edge set 𝐸, the open neighbourhood of the vertex 𝑥 (denoted 

by 𝑁(𝑥)) is the set of all those vertices that are adjacent to 𝑥 and the closed neighbourhood of the 

vertex 𝑥 (denoted by 𝑁[𝑥]) is the union of the open neighbourhood of 𝑥 and {𝑥}.  

For a function 𝑔 defined on the vertex set 𝑉 and having values in the set {0, 1,2}, the weight of a 

vertex 𝑥 is 𝑔(𝑥). The weight of a subset 𝑋 of 𝑉 is denoted by 𝑔(𝑋) and is defined as the sum of the 

weights of all the vertices in 𝑋.  

For 𝑙 ∈ {0,1,2}, we say that a vertex 𝑥 ∈ 𝑉𝑙 if and only if the weight of 𝑥 under 𝑔 is 𝑙. 

In view of the one-to-one correspondence between the functions 𝑔 defined on 𝑉 and having values 

in the set {0,1,2} and the sets ({𝑉𝑙}: 𝑙 = 0,1,2), we can write 𝑔 =  (𝑉0, 𝑉1, 𝑉2). 



Communications on Applied Nonlinear Analysis 

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The function 𝑔 is called a Roman Dominating Function (𝑅𝐷𝐹) [2] , any vertex 𝑎 of Γ having 

weight 0 has at least one vertex with weight 2 in its open neighbourhood. The number 𝛾𝑅(Γ) is the 

minimum of the weights of all 𝑅𝐷𝐹𝑠 defined on Γ. The 𝑅𝐷𝐹 𝑔 with weight 𝛾𝑅(Γ) is called a 𝛾𝑅 − 

function on Γ and the number 𝛾𝑅(Γ) itself is called the 𝛾𝑅 − value of Γ. Many researchers have 

investigated the Roman Dominating Number of graphs [1, 4, 5, 7, 8, 9].  

  A weak Roman Dominating Function 𝑔 (𝑊𝑅𝐷𝐹) [3] is a function such that for any vertex 𝑥 

having positive weight, there is a vertex 𝑦 in the open neighbourhood of 𝑥 having positive weight and 

under the function ℎ defined on 𝑉 having values in the set {0,1,2} defined by ℎ(𝑥) = 1, ℎ(𝑦) =

𝑔(𝑦) − 1, ℎ(𝑧) = 𝑔(𝑧), 𝑧 ∈ 𝑉 − {𝑥, 𝑦}, there are no undefended vertices in Γ. The number 𝛾𝑟(Γ), the 

minimum of the weights of all the 𝑊𝑅𝐷𝐹𝑠 defined on Γ is called the weak Roman Domination Number 

of Γ. The 𝑊𝑅𝐷𝐹 whose weight is 𝛾𝑟(Γ) is called a 𝛾𝑟 − function defined on Γ and 𝛾𝑟(Γ) is called the 

𝛾𝑟 − value of Γ. 𝑊𝑅𝐷𝐹𝑠 are marginally compromising but more accommodative versions of 𝑅𝐷𝐹𝑠. 

Researchers have extensively worked on the parameter weak Roman Domination Number of graphs 

[6, 10, 11, 12]. 

With respect to a graph Γ, we say that an edge 𝑥 ∈ 𝐸− if and only if its removal will reduce the 

𝛾𝑟 − value of Γ. Similarly, we say that an edge 𝑥 ∈ 𝐸+ if and only if its removal will increase the 𝛾𝑟 − 

value of Γ and an edge 𝑥 ∈ 𝐸0 if and only if its removal will leave unaltered the 𝛾𝑟 − value of Γ. The 

behaviour of an edge to belong to 𝐸−, 𝐸+ or 𝐸0 is called the changing and unchanging behaviour of 

the edge. This paper is dedicated to study the changing and unchanging behaviour of an edge with 

respect to a 𝑊𝑅𝐷𝐹.  

 

Definition 1.1. [12] Under a 𝑊𝑅𝐷𝐹, a vertex 𝑎 ∈ 𝑉 with weight 0 is said to be dependent on another 

vertex 𝑏 ∈ 𝑉 of positive weight, if the increase in the weight of 𝑎 by 1 and the simultaneous decrease 

of the weight of 𝑏 by 1, will not create an undefended vertex in 𝑉. We then write, 𝑎 ∈ 𝐷𝐺(𝑏).  

 

Notation 1.1. [12] 𝐻𝑥 = (𝑉1 ∪ 𝑉2) − {𝑥}, �̅�(𝑢) = (𝑁(𝑢) ∩ 𝑉0) − ⋃ 𝐷Γ(𝑥)𝑥∈𝐻 . 

 

Definition 1.2. [12] Let 𝑢 ∈ 𝑉. A (𝑢: 1) − set in Γ is a set 𝑆 ⊆ 𝑁(𝑢) such that for any 𝑥 ∈ 𝑆, 𝑆 −

𝑁(𝐻𝑢) ⊆ 𝑁[𝑥].  

 

Definition 1.3. [12] Let 𝑢 ∈ 𝑉. A (𝑢: 1) − set in Γ is a set 𝑆 ⊆ 𝑁(𝑢) that is not a (𝑢: 1) − set. 

 

2.  Categorizing an edge for membership in 𝑬−, 𝑬+  and 𝑬𝟎 

We start by stating two trivial observations. 

Observation 2.1. 𝐸− =  ∅. 

Observation 2.2. If 𝑥𝑦 is an edge of Γ and if {𝑥, 𝑦} ⊆  𝑉0 or if {𝑥, 𝑦} ⊆  𝑉1 ∪ 𝑉2, then 𝑥𝑦 ∈ 𝐸0. 

Theorem 2.1. Let 𝑥𝑦 be an edge of the graph Γ such that 𝑥 has positive weight and 𝑦 has weight 0 

under a 𝑊𝑅𝐷𝐹 𝑔. Then, the following conditions imply and are implied by the membership of 𝑥 in 

𝐸0.  

1. 𝑦 ∈ �̅�(𝑥) and 

a. 𝑥 ∈ 𝑉2 and �̅�(𝑥) − {𝑦} is a (𝑥: 1) − set in Γ or 



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b. 𝑥 ∈ 𝑉1, 𝑥 ∈ 𝐷Γ(𝜔), for some 𝜔 ∈ 𝑉1 ∪ 𝑉2 and for no vertex 𝑧 ∈ 𝑁(𝑥) ∩ 𝑉0 having 

𝑁(𝑧) ∩ 𝑉2 = ∅ and for every 𝑡 ∈ (𝑁(𝑧) − {𝑥}) ∩ 𝑉1, we have �̅�(𝑡) ∪ {𝑦} is a (𝑡: 2) − 

set in Γ. 

2. 𝑦 ∉ �̅�(𝑥) and 

c. 𝑦 ∈ 𝐷Γ(𝑟) for some 𝑟 ∈ 𝑉2 or if 𝑁(𝑦) ∩ 𝑉2 = ∅, 𝑦 ∈ 𝐷Γ(𝑟), for some 𝑟 ∈ 𝑉1 where 

|𝑁(𝑦) ∩ 𝑉1| > 2  if |�̅�(𝑟) ∪ {𝑦}| > 1 and �̅�(𝑟) ∪ {𝑦} does not induce a clique and 

�̅�(𝑥) − {𝑦} is a (𝑥: 𝑖) − set in Γ where 𝑔(𝑥) = 𝑖 or  

d. 𝑟 ∈ 𝑉1, �̅�(𝑟) ∪ {𝑦} is a (𝑟: 2) − set and we can find 𝑚 ∈ 𝑉1⋃𝑉2 such that 𝑥 ∈ 𝐷Γ(𝑚). 

 

Proof. Assume that 𝑦 ∈ �̅�(𝑥), 𝑥 ∈ 𝑉2. Let �̅�(𝑥) − {𝑦} be a (𝑥: 1) − set in Γ. Consider the function 𝑔′ 

defined on 𝑉 and having values in {0,1,2} by 𝑔′(𝑦) = 1, 𝑔′(𝑥) = 1 and 𝑔′(𝑧) = 𝑔(𝑧) for all 𝑧 ∈ 𝑉 −

{𝑥, 𝑦}. The function 𝑔′ is a 𝑊𝑅𝐷𝐹 on Γ − 𝑒 having weight 𝛾𝑟(Γ). Since 𝑒 ∉ 𝐸−, 𝑔′ is a 𝛾𝑟 − function 

defined on Γ − 𝑒. Therefore, 𝛾𝑟(Γ − 𝑒) = 𝛾𝑟(Γ). So, 𝑒 ∈ 𝐸0.  

Assume now that, 𝑥 ∈ 𝑉1. If 𝑥 ∈ 𝐷𝐺(𝑤) for some 𝑤 ∈ 𝑉1⋃𝑉2 then there exists no 𝑧 ∈ 𝑉0⋂𝑁(𝑥) 

with 𝑁(𝑧) ∩ 𝑉2 = ∅ and for every 𝑡 ∈ 𝑁(𝑧)⋂𝑉1, if we have �̅�(𝑡) ∪ {𝑦} is a (𝑡: 2) − set in Γ, then the 

function 𝑔′ defined on 𝑉 having values in {0,1,2} defined by 𝑔′(𝑦) = 1, 𝑔′(𝑥) = 0, 𝑔′(𝑠) = 𝑔(𝑠) for 

all 𝑠 ∈ 𝑉 − {𝑥, 𝑦} is a 𝑊𝑅𝐷𝐹 defined on Γ − 𝑒 having weight 𝛾𝑟(Γ). Since 𝑒 ∉ 𝐸− we have 𝑔′ is a 

𝛾𝑟 − function defined on Γ − 𝑒. Therefore, 𝛾𝑟(Γ − 𝑒) = 𝛾𝑟(Γ). So, 𝑒 ∈ 𝐸0.  

 Assume now that 𝑦 ∉ �̅�(𝑥). Assume that there is some 𝑟 ∈ 𝑉2. Then, 𝑒 ∈ 𝐸0. Let 𝑁(𝑦)⋂𝑉2 =

∅, 𝑦 ∈ �̅�(𝑟) for some 𝑟 ∈ 𝑉1 where |𝑁(𝑦) ∩ 𝑉1| > 2 if |�̅�(𝑟) ∪ {𝑦}| > 1, �̅�(𝑟) ∪ {𝑦} does not induce 

a clique in such a manner that �̅�(𝑥) − {𝑦} happens to be a (𝑢: 𝑖) − set, where 𝑔(𝑥) = 𝑖, 𝑖 = 1, 2. In 

this case, 𝑔 is a 𝑊𝑅𝐷𝐹 defined on Γ − 𝑒. Again since 𝑒 ∉ 𝐸−, we have that 𝑔′ is a 𝛾𝑟 − function 

defined on Γ − 𝑒. Therefore, 𝛾𝑟(Γ − 𝑒) = 𝛾𝑟(Γ). Consequently, 𝑒 ∈ 𝐸0. 

 If 𝑟 ∈ 𝑉1, and �̅�(𝑟) ∪ {𝑦} is a (𝑟: 2) − set in Γ and if we can find 𝑚 ∈ 𝑉1 ∪ 𝑉2 in such a manner 

that 𝑥 ∈ 𝐷𝐺(𝑚), then the function 𝑔′ defined on 𝑉 having values in {0,1,2} so defined that 𝑔′(𝑟) =

2, 𝑔′(𝑥) = 0, 𝑔′(𝑧) = 𝑔(𝑧) when 𝑧 ∈ 𝑉 − {𝑟, 𝑥} will be a 𝑊𝑅𝐷𝐹 defined on Γ − 𝑒 having 𝛾𝑟(Γ) as its 

weight. Arguing as before, we have 𝑒 ∈ 𝐸0. 

 Conversely, let 𝑒 ∈ 𝐸0. If 𝑦 ∈ �̅�(𝑥), then 𝑒 ∈ 𝐸+ if none of the conditions a or b is satisfied. 

Similarly, if 𝑦 ∈ �̅�(𝑥), then 𝑒 ∈ 𝐸+ if none of the conditions c or d is satisfied. 

 This proves the theorem completely.                                                                                        

 

Theorem 2.2.  For any graph Γ having an edge 𝑥𝑦 (= 𝑒), 𝑒 ∈ 𝐸+ if and only if 𝑒 ∉ 𝐸0. 

 

Proof. The result is a consequence of the facts, 𝐸 = 𝐸− ∪ 𝐸0 ∪ 𝐸+, 𝐸− = ∅ and 𝐸0 ∩ 𝐸+ = ∅.                                                                                                                                                                        

 

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