Communications on Applied Nonlinear Analysis 

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Changing and Unchanging Secure Integer Domination in Graphs 

𝑮𝒐𝒘𝒕𝒉𝒂𝒎 𝑷𝒓𝒊𝒚𝒂 𝑳∗𝟏,    𝑽𝒆𝒏𝒌𝒂𝒕𝒆𝒔𝒉 𝑲 𝑨𝟐 

1 Department of Mathematics, Alliance University, Bengaluru, Karnataka 5621064, India. 

2 School of Advanced Computing, Alliance University, Bengaluru, Karnataka 5621064, India. 

gowthampriya.28@gmail.com 

Article History: 

Received: 12-11-2024 

Revised:24-12-2024 

Accepted:09-01-2025 

 

Abstract:  

An Integer dominating function on a graph G is a function f : V (G) → W such that for 

every vertex 

 v ∈ V (G), ∑ (𝑁[𝑣]) ≥ 𝑘𝑣 ∈ 𝑉(𝐺) . For any function f : V (G) → W and any pair of 

adjacent vertices with f(v) = 0 and u > 0, the function guv is defined by 𝑔𝑢𝑣 (l) = 1, 

𝑔𝑢𝑣 (l) = f(u) − 1 and 𝑔𝑢𝑣 (l) = f(l) if 𝑙 ∈  𝑉 − {𝑢, 𝑣}. A secure integer dominating 

function on a graph G is defined as an integer dominating function g which satisfies 

that for every vertex v with f(v) = 0, a neighbour u with f(u) > 0 such that 𝑔𝑢𝑣 is an 

integer dominating function. The weight of f is w(f) = ∑ 𝑓(𝑣)𝑣 ∈ 𝑉(𝐺) . Minimum weight 

among all the secure integer dominating function on G is secure integer domination 

number on G. This paper is devoted to initiating the study of SIDF of a graph. In 

particular, we have studied the changing and unchanging behavior of the graphs. 

 

Objectives: We propose a novel generalization of domination, which incorporates 

additional security and broader applicability. This refined framework offers new 

possibilities for research and practical implementation. 

Keywords: Domination, Secure domination, Integer domination, changing and 

unchanging domination. 

1. Introduction 

The study of domination can be traced back to 1862, when de Jaenish attempted to determine 

the minimum number of queens required to cover the n ∗ n chess board. The study of 

domination in graphs was further developed in 1958 by C. Berge and O. Ore in 1962 [6]. One 

of the variation of domination is Secure domination in graphs. This was studied and introduced 

by E. J. Cockayne et.al [4]. Secure dominating sets can be applied as protection strategies by 

minimizing the number of guards to secure a system so as to be cost effective as possible. For 

the general concepts not mentioned, the readers may be referred to [11]. 

A graph G is a pair (𝑉(𝐺), 𝐸(𝐺)), where V(G) is a finite nonempty set called the vertex set of 

G and E(G) is a set of unordered pairs xy of distinct elements from V(G) called the edge set of 

G. The elements of V(G) are called vertices. The order of G is denoted by n =| V(G) | and the 

size of G by m =| E(G) |. A subset 𝐻 ∈ 𝑉(𝐺) the subgraph induced by H is the graph G[H] 

with vertex set H and edge set {xy ∈ E(G) | x, y ∈ H}. We write Kn for complete graph of order 

n, Km,n for complete bipartite graph with partite sets of order n and m, Pn for the path on n 

vertices and Cn for the cycle of length n. A star is the graph 𝑆1,𝑟 where r ≤ 1. For any vertex v 

∈ V(G) open neighbourhood of v is the set N(v) = {u ∈ V | uv ∈ E} and the closed neighbourhood 



Communications on Applied Nonlinear Analysis 

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is the set N[v] = N(v) ∪ v. The private neighbor set of a v ∈ V(G) with respect to a set D, 

denoted by pn[v,D] is N[v]N[Sv] and each u ∈ pn[v,D] is called a private neighbor of v(G) 

with respect to D. 

O. Ore introduced the concept of Domination theory. A set D ∈ V(G) is a dominating set if 

every vertex in V(G)is adjacent to at least one vertex in D. The domination number γ(G) is the 

minimum cardinality of a dominating set of G. Domination theory has wide application in real 

life. Various kinds of domination have been studied in recent times, one of the variation is 

Integer domination. Domke et al.[3] introduced the concept of Integer k-domination and further 

integer domination on Vizing’s conjecture was studied by Bresar et al.[2]. A function f : V (G) 

→ W (W is the whole number) is called integer k-dominating function if the sum of functional 

value over any closed neighbor is at least k. The weight of integer k-dominating function is the 

value of F(v) =∑ 𝑓(𝑣)𝑣 ∈𝑉(𝐺) . The minimum weight of integer {k}− dominating function is 

denoted as 𝛾𝑘(G) and is called integer domination number. Note that, when k = 1 its a general 

domination. 

In this article, we introduce secure integer domination and study about changing and 

unchanging behavior of graph. 

A function g : V (G) → {0, 1, 2, .., k} is called secure integer dominating function (SIDF) of G 

if it satisfies the following: 

1) ∀ 𝑦 ∈ 𝑉(𝐺), ∑ 𝑔(𝑁[𝑦]) ≥ 𝑘.𝑣∈𝑉(𝐺)  

2) ∀ 𝑦 ∈  𝑉0 ∃ 𝑧 ∈  𝑁(𝑦) −  𝑉0 such that 𝑔𝑦𝑧 is an integer dominating function on G. 

 

The secure integer dominating function has a weight equal to the value of w(y) =∑ 𝑔(𝑦)𝑦 ∈𝑉(𝐺) . 

The minimum weight of secure integer dominating function is denoted by 𝛾𝑘
𝑠(𝐺) is called 

secure integer domination number (SIDN). 

For a SIDF g, let 𝑉𝑖
𝑔

=  { v ∈ V(G) : f (v) = i} for i = 0, 1, 2,..., k. Since these k sets 

determine g, we can equivalently write 𝑔 = (𝑉0
𝑔

, 𝑉1
𝑔

, 𝑉2
𝑔

, . . . 𝑉𝑘
𝑔

). We examine the 

effects on the domination number when the graph is modified by deleting a vertex or deleting 

or adding an edge. Let G-v denote the graph formed by removing vertex v and G − e denote 

the graph formed by removing edge e from G. We use acronyms to denote these classes (V 

represents vertex; E: edge; R: removal; A: addition). 

Note that, there are six class of subgraphs obtainted. The following are the class of graphs. 

• 𝛾𝑘
𝑠(𝐺 − 𝑣)  ≠   𝛾𝑘

𝑠(𝐺) for all 𝑣 ∈  𝑉(𝐺).  (CVR) 

• 𝛾𝑘
𝑠(𝐺 − 𝑣)  =   𝛾𝑘

𝑠(𝐺) for all 𝑣 ∈  𝑉(𝐺).  (UVR) 

• 𝛾𝑘
𝑠(𝐺 − 𝑒)  ≠   𝛾𝑘

𝑠(𝐺) for all 𝑒 ∈  𝐸(𝐺).  (CER) 

• 𝛾𝑘
𝑠(𝐺 − 𝑒)  =   𝛾𝑘

𝑠(𝐺) for all 𝑒 ∈  𝐸(𝐺).  (UER) 

• 𝛾𝑘
𝑠(𝐺 + 𝑒)  ≠   𝛾𝑘

𝑠(𝐺) for all 𝑒 ∈  𝐸(𝐺).  (CEA) 

• 𝛾𝑘
𝑠(𝐺 − 𝑒)  =   𝛾𝑘

𝑠(𝐺) for all 𝑒 ∈  𝐸(𝐺).  (UEA) 

 



Communications on Applied Nonlinear Analysis 

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Here we examine two cases that is, 𝛾𝑘
𝑠(𝐺 − 𝑣)  ≠   𝛾𝑘

𝑠(𝐺) for all 𝑣 ∈  𝑉(𝐺) . We partition the 

vertex set of G into three sets according to their removal affects 𝛾𝑘
𝑠(𝐺). Let 𝑉(𝐺) = {𝑉0 ∪

 𝑉+ ∪ 𝑉−} where 𝑉0 =  { 𝑢 ∈  𝑉(𝐺): 𝛾𝑘
𝑠(𝐺 − 𝑣) = 𝛾𝑘

𝑠(𝐺)}, 𝑉+ = {𝑢 ∈  𝑉(𝐺): 𝛾𝑘
𝑠(𝐺 − 𝑣) >

𝛾𝑘
𝑠(𝐺)} and 𝑉− =  { 𝑢 ∈  𝑉(𝐺): 𝛾𝑘

𝑠(𝐺 − 𝑣) < 𝛾𝑘
𝑠(𝐺)}. Similarly, the edge set can be partitioned 

into 𝐸0 =  { 𝑥𝑦 ∈  𝐸(𝐺) ∶  𝛾𝑘
𝑠(𝐺 − 𝑥𝑦) = 𝛾𝑘

𝑠(𝐺)} and 𝐸+ =  { 𝑥𝑦 ∈  𝐸(𝐺): 𝛾𝑘
𝑠(𝐺 − 𝑥𝑦) >

𝛾𝑘
𝑠(𝐺)}. 

2. Results: 

Theorem 2.1. For every vertex v in Kn, 𝛾𝑘
𝑠(𝐺 − 𝑣) = 𝛾𝑘

𝑠(𝐺). 

Proof. Let G = Kn be a complete graph with n vertices. On contrary, assume that  𝛾𝑘
𝑠(𝐺 − 𝑣) ≠

  𝛾𝑘
𝑠(𝐺). Since every vertex is of degree 𝑛 − 1. 𝐺 − 𝑣 will be a graph with 𝑛 − 1 vertices and 

degree 𝑛 − 2, which will be a complete graph with Kn−1. Removal of vertex in G, does not 

affect secure domination number. Therefore, our assumption is wrong. 

Theorem 2.2.  Let 𝐺 = 𝑆1,𝑟 and v = Δ(G) − 1, then 𝛾𝑘
𝑠(𝐺 − 𝑣) >  𝛾𝑘

𝑠(𝐺).  

Proof. Let G = S1,r be a star graph, where r = {𝑟1, 𝑟2, . . . 𝑟𝑛−1}. On contrary, assume that 𝛾𝑘
𝑠(𝐺 −

𝑣) <  𝛾𝑘
𝑠(𝐺). Let v be a vertex with degree 𝑛 − 1, removal of vertex v, will disconnect the 

graph. We obtain a graph with 𝑛 − 1  vertices and 𝑛 − 1  component, which increase SIDN. 

Therefore, our assumption is wrong. 

Theorem 2.3. A vertex v in 𝑉− iff pn[v,D] = v for some 𝛾𝑘
𝑠 set D containing v. 

Proof. Let v ∈  𝑉−and S be a 𝛾𝑘
𝑠 set of G − v. Then D = S ∪ {v} is a 𝛾𝑘

𝑠  set of G. If S contains 

a vertex of N(v), then S is a dominating set of G, which contradicts our assumption. Thus, 

pn[v,D] = {v}. Conversely, D − {v} dominates 𝐺 − 𝑣, therefore v ∈ 𝑉−. 

Theorem 2.4. For any tree T with n ≥ 2, there exists a vertex v ∈ V (G) such that 𝛾𝑘
𝑠(𝑇 − 𝑣) =

𝛾𝑘
𝑠(𝐺). 

Proof. Assume that T has atleast one vertex v with deg(v) ≥ 2 that is adjacent to atleast one 

endvertex and at most one non endvertex. If v is adjacent to two or more endvertices 𝑟1 and 𝑟2, 

then v is in every 𝛾𝑘
𝑠 set for T and 𝛾𝑘

𝑠(𝑇 − 𝑟1) = 𝛾𝑘
𝑠(𝑇). If not, then v is adjacent to one endvertex 

r and deg = 2. Let 𝑇′= T − v − r. For any graph G, if deg(r) − 1, then 𝛾𝑘
𝑠(𝐺 − 𝑟) ≤ 𝛾𝑘

𝑠(𝐺). 

Hence 𝛾𝑘
𝑠(𝑇′) ≤ 𝛾𝑘

𝑠(𝑇). However, 𝛾𝑘
𝑠(𝑇′) ≤ 𝛾𝑘

𝑠(𝑇) − 1. If 𝛾𝑘
𝑠(𝑇′) = 𝛾𝑘

𝑠(𝑇) − 1, then 𝛾𝑘
𝑠(𝑇) ≤

𝛾𝑘
𝑠(𝑇 − 𝑣). Otherwise, 𝛾𝑘

𝑠(𝑇′) = 𝛾𝑘
𝑠(𝑇) = 𝛾𝑘

𝑠(𝑇 − 𝑣). 

3. Conclusions 

 In this article, we have studies introduced the concept of secure integer domination in graphs. 

We have observed the changing and unchanging behavior of secure integer domination in some 

graphs. To generalize, the changing and unchanging secure integer domination in graph is 

open. 

 Funding 

This research received no specific grant from public, commercial, or not-for-profit funding 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

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agencies. 

Acknowledgments 

The authors wish to thank the management of Alliance University, Bengaluru, Karnataka 

562106, India., for their continuous support and encouragement to carry out this research work. 

Conflicts of Interest  

The authors declare that none of the work reported in this study could have been influenced by 

any known competing financial interests or personal relationships. 

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