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Analyzing Detour Distance and Domination Number in Special 

Graph Classes 

N. Jeyasree1 and Dr S. Chelliah2 

1Research Scholar, Register Number: 21121072092002, Department of Mathematics, The 

M.D.T Hindu College, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, 

Tirunelveli, India 

jeyasree1960@gmail.com 

2Associate Professor, Department of Mathematics, The M.D.T Hindu College, Affiliated to 

Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli, India 

kscmdt@gmail.com 

Article History:  

 Received:01/10/2024 

Revised: 06/11/2024 

Accepted: 30/12/2024 

Abstract:   

In this paper, we investigate the relationship between detour distance and 

domination number within various special classes of graphs. The detour 

distance between two vertices is defined as the length of the longest path 

connecting them, and it provides an alternative metric to the traditional 

geodesic distance in graph theory. By integrating this concept with 

domination theory, we introduce new structural measures that capture the 

extended influence of vertices beyond their immediate neighborhoods. We 

define and analyze the detour domination number, denoted as γDD(G), which 

represents the minimum cardinality of a set of vertices such that every vertex 

in the graph lies on a detour path from at least one dominating vertex. We 

examine the behavior of this parameter in specific graph classes, including 

complete graphs, paths, cycles, trees, and grid graphs, deriving exact values 

or bounds where applicable. Additionally, we study the upper detour 

domination number, investigate its extremal properties, and explore how it 

contrasts with traditional domination metrics. We also characterize graph 

classes where the detour domination number equals the classical domination 

number and where it significantly diverges. The results contribute to a deeper 

understanding of detour-based influence in graphs, offering new perspectives 

for applications in network resilience, transport systems, and distributed 

computing, where long-range reachability and control are essential. 

Keywords: Detour Distance, Detour Dominating Set, Domination Number, 

Upper Detour Domination, Forcing Set in Graphs, Special Graph Classes. 

INTRODUCTION 

Graph theory, a foundational discipline in discrete mathematics, has evolved to become a 

powerful tool for modeling a wide variety of systems in science, engineering, social networks, 

computer science, and operations research. One of the most widely studied problems in graph 

theory is the domination problem, which seeks a subset of vertices that exerts influence or 

control over the entire graph. Traditional domination theory focuses on minimum sets of 

mailto:jeyasree1960@gmail.com
mailto:kscmdt@gmail.com


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vertices that are adjacent to all other vertices either directly or indirectly. However, in many 

real-world networks—such as communication systems, transportation grids, biological 

systems, and social networks—the shortest-path assumption is often unrealistic. This motivates 

the need for domination models that incorporate longest-path interactions, leading to the 

development of detour-based domination concepts. 

This work introduces and elaborates upon new graph invariants based on detour domination 

and detour distances. First, we formalize the detour dominating degree set, which includes sets 

that not only detourdominate the graph but are evaluated based on degree-based metrics of 

influence. We then introduce the upper detour dominating degree set, which seeks to maximize 

the local domination measure based on vertex degrees involved in the detour structure. Such 

sets provide insight into the maximum structural influence that can be exerted through long-

distance interactions in the graph. The associated invariant, the upper detour domination 

number 𝛾𝑑𝑑
+ (𝐺), captures the maximum weighted detour reachability. 

PRELIMINARIES 

Definition 1: Detour Distance 

Let 𝐺 = (𝑉, 𝐸) be a connected graph and let 𝑢, 𝑣 ∈ 𝑉(𝐺). 

The detour distance 𝐷(𝑢, 𝑣) is defined as the length of the longest 𝑢 − 𝑣 path in the graph 𝐺. 

Formally, 

𝐷(𝑢, 𝑣) = max{ length (𝑃) ∣ 𝑃 is a 𝑢 − 𝑣 path in 𝐺} 

This measure contrasts with the usual distance 𝑑(𝑢, 𝑣), which considers the shortest path. The 

detour distance captures long-range connectivity between vertices and is particularly relevant 

in resilience and influence spread models. 

Definition 2: Detour Dominating Set 

A subset 𝑆 ⊆ 𝑉(𝐺) is called a detour dominating set if for every vertex 𝑣 ∈ 𝑉(𝐺), there exists 

a vertex 𝑢 ∈ 𝑆 such that 𝑣 lies on a detour path originating from 𝑢. That is, 

∀𝑣 ∈ 𝑉(𝐺), ∃𝑢 ∈ 𝑆 such that 𝑣 ∈ 𝑉(𝑃𝑢𝑣), where 𝑃𝑢𝑣 is a detour path from 𝑢 to 𝑣 

This definition implies that 𝑆 collectively "controls" the graph via its longest internal routes, 

not just the immediate neighborhoods. 

Definition 3: Detour Dominating Degree Set (𝜸𝒅𝒅(𝑮) ) 

A detour dominating degree set 𝑆 ⊆ 𝑉(𝐺) is a detour dominating set with minimum 

cardinality. The detour domination degree number, denoted by 𝛾𝑑𝑑(𝐺), is defined as: 

𝛾𝑑𝑑(𝐺) = min{|𝑆| ∣ 𝑆 ⊆ 𝑉(𝐺), 𝑆 is a detour dominating set } 

This number reflects the minimum number of vertices required to detour-dominate the entire 

graph. It serves as a generalization of the classical domination number under the lens of 

maximum connectivity. 

Definition 4: Upper Detour Dominating Degree Set ( 𝜸𝒅𝒅
+ (𝑮) ) 

Let 𝐺 be a graph and let 𝑆 ⊆ 𝑉(𝐺) be a detour dominating set. The set 𝑆 is called an upper 

detour dominating degree set if it maximizes a structural parameter called the local detour 



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domination degree, denoted 𝑙𝑑(𝑠). The upper detour domination degree number 𝛾𝑑𝑑
+ (𝐺) is 

given by: 

𝛾𝑑𝑑
+ (𝑢, 𝑣) = max{𝑙𝑑(𝑠)} = max {deg(𝑢) + deg(𝑣) + ∑  

𝑤∈𝑁(𝑢,𝑣)

 deg(𝑤)} 

where: 

• 𝑢, 𝑣 ∈ 𝑆 

• 𝑁(𝑢, 𝑣) is a neighborhood of vertices lying on the detour path from 𝑢 to 𝑣 

• deg(𝑥) denotes the degree of vertex 𝑥 

This definition captures the strongest detour influence a subset can exert in terms of degree 

and detour structure. 

THEOREMS 

Theorem 1: Detour Domination Number in Complete Graphs 

Theorem: 

Let 𝐾𝑛 − (𝑉, 𝐸) be a complete graph on 𝑛 ≥ 2 vertices. Then the detour domination number 

𝛾𝐷(𝐾𝑛) is 1, i.e., 

𝛾𝐷(𝐾𝑛) − 1 

Proof: 

In a complete graph 𝐾𝑛, every pair of distinct vertices 𝑢, 𝑣 ∈ 𝑉 is connected by a unique 

edge, so: 

𝑑(𝑢, 𝑣) − 1 ∀𝑢 ≠ 𝑣 

However, the detour distance 𝐷(𝑢, 𝑣), defined as the length of the longest possible path 

between 𝑢 and 𝑣 without repeating any vertices, is: 

𝐷(𝑢, 𝑣) − 𝑛 − 1 ∀𝑢 ≠ 𝑣 

since the longest path between any two vertices in 𝐾𝑛 must visit all other 𝑛 − 2 vertices 

exactly once before reaching 𝑣, utilizing all 𝑛 − 1 vertices (including 𝑢 and 𝑣 ). 

Let 𝑆 ⊆ 𝑉 be a detour dominating set, i.e., every vertex 𝑣 ∈ 𝑉 ∖ 𝑆 must lie on a detour path 

that includes some 𝑢 ∈ 𝑆. Due to the complete connectivity of 𝐾𝑛, from any chosen vertex 

𝑢 ∈ 𝑉, there exists a detour path of length 𝑛 − 1 that visits every other vertex 𝑣 ∈ 𝑉 ∖ {𝑢}. 

Hence, one vertex suffices to detour dominate the entire graph. 

Therefore: 

𝛾𝐷(𝐾𝑛) − 1 

To formalize this in matrix terms, let 𝐴 ∈ ℝ𝑛×𝑛 be the adjacency matrix of 𝐾𝑛, defined by: 

𝐴 − 𝐽𝑛 − 𝐼𝑛 

where 𝐽𝑛 is the all-ones matrix and 𝐼𝑛 is the identity matrix. The entries of 𝐴 are given by: 



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𝐴𝑖𝑗 = {
1  if 𝑖 ≠ 𝑗
0  if 𝑖 − 𝑗

 

Let 𝑥 − (𝑥1, 𝑥2, … , 𝑥𝑛)𝑇 be the characteristic vector of the detour dominating set 𝑆 ⊆ 𝑉, 

where: 

𝑥𝑖 − {
1  if 𝑣𝑖 ∈ 𝑆
0  otherwise 

 

Then the detour domination constraint requires that for each 𝑗 − 1,2, … , 𝑛, there must exist at 

least one 𝑖 ≠ 𝑗 such that 𝑥𝑖 − 1. This is equivalent to: 

∑  

𝑛

𝑖=1
𝑖≠𝑗

𝑥𝑖 ≥ 1 

However, since 𝐾𝑛 allows for each vertex to lie on a detour path from any other vertex, and 

since every vertex is adjacent to all others, a single vertex suffices for detour domination. 

Thus, the integer program becomes: 

 Minimize ∑  

𝑛

𝑖=1

 𝑥𝑖

 Subject to ∑  

𝑛

𝑖=1
𝑖≠𝑗

 𝑥𝑖 ≥ 1 for all 𝑗 − 1,2, … , 𝑛

𝑥𝑖 ∈ {0,1}, 𝑖 − 1,2, … , 𝑛

 

Setting any one 𝑥𝑘 − 1 and all others 𝑥𝑖 − 0 for 𝑖 ≠ 𝑘 satisfies all constraints, yielding the 

minimum value: 

𝛾𝐷(𝐾𝑛) − ∑  

𝑛

𝑖=1

𝑥𝑖 − 1 

This completes the proof. 

 

Theorem 2: Detour Domination Number in Path Graphs 𝑃𝑛 

Theorem: 

Let 𝑃𝑛 − (𝑉, 𝐸) denote a simple undirected path graph on 𝑛 ≥ 1 vertices with consecutive 

edges. Then the detour domination number 𝛾𝐷(𝑃𝑛) satisfies: 

𝛾𝐷(𝑃𝑛) − ⌈
𝑛

3
⌉ 

Proof: 

Let 𝑉 − {𝑣1, 𝑣2, … , 𝑣𝑛}, and let edges 𝐸 − {(𝑣𝑖 , 𝑣𝑖+1) ∣ 1 ≤ 𝑖 < 𝑛} connect consecutive 



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vertices. Then 𝑃𝑛 is a path of length 𝑛 − 1. 

The distance 𝑑(𝑣𝑖, 𝑣𝑗) between any two vertices 𝑣𝑖 and 𝑣𝑗 is: 

𝑑(𝑣𝑖 , 𝑣𝑗) − |𝑖 − 𝑗| 

The detour distance 𝐷(𝑣𝑖 , 𝑣𝑗), defined as the length of the longest simple path between 𝑣𝑖 and 

𝑣𝑗, equals: 

𝐷(𝑣𝑖 , 𝑣𝑗) − 𝑛 − 1,  for (𝑣𝑖 , 𝑣𝑗) − (𝑣1, 𝑣𝑛) 

Let 𝑆 ⊆ 𝑉 be a detour dominating set such that for every vertex 𝑣𝑘 ∈ 𝑉, there exists 𝑣𝑖 ∈ 𝑆 

for which 𝑣𝑘 lies on a detour path (of length 𝑛 − 1 ) beginning at 𝑣𝑖. That is: 

∀𝑣𝑘 ∈ 𝑉, ∃𝑣𝑖 ∈ 𝑆 such that 𝑣𝑘 ∈ Pathdetour (𝑣𝑖) 

We construct such a set 𝑆 by selecting every third vertex, i.e., 

𝑆 − {𝑣𝑖 ∈ 𝑉 ∣ 𝑖 ≡ 2(mod3)} 

Each such 𝑣𝑖 ∈ 𝑆 can detour-dominate its neighbors: 

𝑁𝐷[𝑣𝑖] − {𝑣𝑖−1, 𝑣𝑖 , 𝑣𝑖+1} ∩ 𝑉 

Thus, the total detour coverage from all selected dominators is: 

⋃  

𝑣𝑖∈𝑆

𝑁𝐷[𝑣𝑖] − 𝑉 

and the size of such a set is: 

|𝑆| = ⌈
𝑛

3
⌉ 

Now, to formulate the problem algebraically, define indicator variables: 

𝑥𝑖 = {
1  if 𝑣𝑖 ∈ 𝑆
0  otherwise 

 for 𝑖 − 1,2, … , 𝑛 

Then, for each vertex 𝑣𝑘 ∈ 𝑉, the domination constraint is: 

𝑥𝑘−1 + 𝑥𝑘 + 𝑥𝑘+1 ≥ 1 ∀1 ≤ 𝑘 ≤ 𝑛 

with boundary conditions handled by setting: 

𝑥0 − 𝑥𝑛+1 − 0 

The optimization problem becomes: 

 Minimize: 𝑧 − ∑  

𝑛

𝑖=1

  𝑥𝑖

 Subject to: 𝑥𝑖−1 + 𝑥𝑖 + 𝑥𝑖+1 ≥ 1, ∀𝑖 − 1, … , 𝑛

𝑥0 − 𝑥𝑛+1 − 0

𝑥𝑖 ∈ {0,1}, ∀𝑖

 



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This is an instance of a binary integer programming problem. 

Thus, the solution gives the detour domination number: 

𝛾𝐷(𝑃𝑛) − min {∑  

𝑛

𝑖=1

  𝑥𝑖 ∣ 𝑥𝑖−1 + 𝑥𝑖 + 𝑥𝑖+1 ≥ 1, 𝑥𝑖 ∈ {0,1}, 𝑖 − 1, … , 𝑛} 

In compact set-theoretic optimization form, the detour domination number can also be written 

as: 

𝛾𝐷(𝑃𝑛) = min
𝑥∈{0,1}a

 {∑  

𝑛

𝑖=1

 𝑥𝑖 ∣ ⋃  

𝑖:𝑥𝑖−1

 𝑁𝐷[𝑖] − 𝑉(𝑃𝑛)} 

where: 

𝑁𝐷[𝑖] − {𝑖 − 1, 𝑖, 𝑖 + 1} ∩ [1, 𝑛] 

This completes the proof. 

 

Theorem 3: Detour Domination Number in Cycle Graphs Cn 

Theorem: 

For the cycle graph 𝐶𝑛 on 𝑛 ≥ 3 vertices, the detour domination number satisfies: 

𝛾𝐷𝐷(𝐶𝑛) − ⌈
𝑛

3
⌉ 

Proof: 

Let 𝐶𝑛 − (𝑉, 𝐸) be the cycle graph with vertex set 𝑉 − {𝑣1, 𝑣2, … , 𝑣𝑛} and edge set defined 

by: 

𝐸 − {(𝑣𝑖 , 𝑣𝑖+1) ∣ 1 ≤ 𝑖 < 𝑛} ∪ {(𝑣𝑛, 𝑣1)} 

The distance between any two vertices 𝑣𝑖 and 𝑣𝑗 on the cycle is given by: 

𝑑(𝑣𝑖 , 𝑣𝑗) − min(|𝑖 − 𝑗|, 𝑛 − |𝑖 − 𝑗|) 

The detour distance (longest simple path) between 𝑣𝑖 and 𝑣𝑗 is: 

𝐷(𝑣𝑖, 𝑣𝑗) − ⌊
𝑛

2
⌋ 

To construct a detour dominating set 𝑆 ⊆ 𝑉, we require that for all 𝑣 ∈ 𝑉, there exists some 

𝑣𝑘 ∈ 𝑆 such that 𝑣 ∈ Pathdetour (𝑣𝑘). Since each detour in the cycle is symmetric and spans 

up to half the cycle, it suffices to place dominators every 3 vertices. 

Construct 𝑆 as: 

𝑆 − {𝑣𝑘 ∈ 𝑉 ∣ 𝑘 ≡ 1(mod3)} 

Using modular arithmetic on indices, each dominator 𝑣𝑘 ∈ 𝑆 covers: 



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𝑁𝐷[𝑣𝑘] − {𝑣𝑘−1, 𝑣𝑘, 𝑣𝑘+1} with indices modulo 𝑛 

Therefore, the union of closed detour neighborhoods satisfies: 

⋃  

𝑣𝑘∈𝑆

𝑁𝐷[𝑣𝑘] − 𝑉 

and the size of such a set is: 

|𝑆| − ⌈
𝑛

3
⌉ 

To express this algebraically, define indicator variables: 

𝑥𝑖 − {
1  if 𝑣𝑖 ∈ 𝑆
0  otherwise 

 ∀𝑖 − 1, … , 𝑛 

The domination constraint for each vertex 𝑣𝑗 ∈ 𝑉 is: 

𝑥𝑗−1 + 𝑥𝑗 + 𝑥𝑗+1 ≥ 1,  for 𝑗 − 1, … , 𝑛 

with cyclic (modular) indexing: 

𝑥0 − 𝑥𝑛, 𝑥𝑛+1 − 𝑥1 

 Minimize: ∑  

𝑛

𝑖=1

  𝑥𝑖

 Subject to: 𝑥𝑖−1 + 𝑥𝑖 + 𝑥𝑖+1 ≥ 1 ∀𝑖 − 1, … , 𝑛

𝑥𝑖 ∈ {0,1} ∀𝑖

 

Thus, 

𝛾𝐷𝐷(𝐶𝑛) − min {∑  

𝑛

𝑖=1

 𝑥𝑖 ∣ 𝑥𝑖−1 + 𝑥𝑖 + 𝑥𝑖+1 ≥ 1, 𝑥𝑖 ∈ {0,1}} − ⌈
𝑛

3
⌉ 

Theorem 4: Detour Domination in Trees via Branch Decomposition 

Theorem: 

Let 𝑇 − (𝑉, 𝐸) be a tree of order 𝑛, rooted at a vertex 𝑟, and let it branch into 𝑘 major paths 

𝐵1, 𝐵2, … , 𝐵𝑘 , with each branch 𝐵𝑖  of height ℎ𝑖. Then, 

𝛾𝐷𝐷(𝑇) ≤ ∑  

𝑘

𝑖=1

⌈
ℎ𝑖

2
⌉ 

Proof: 

In a tree, the path between any two vertices is unique. Let 𝑑(𝑢, 𝑣) denote the number of 

edges in the unique 𝑢 → 𝑣 path. Then, 

𝐷(𝑢, 𝑣) − 𝑑(𝑢, 𝑣), ∀𝑢, 𝑣 ∈ 𝑉(𝑇) 

Let the root be 𝑟, and decompose the tree into disjoint branches from 𝑟 as: 



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𝑇 − ⋃  

𝑘

𝑖=1

𝐵𝑖  

Each branch 𝐵𝑖  is a path of height ℎ𝑖. To detour-dominate each 𝐵𝑖 , we select vertices at 

intervals of 2 along the path from root to leaves. Thus, each branch requires: 

|𝑆𝑖| = ⌈
ℎ𝑖

2
⌉  dominators  

Summing over all branches: 

𝛾𝐷𝐷(𝑇) ≤ ∑  

𝑘

𝑖=1

⌈
ℎ𝑖

2
⌉ 

Define binary variables 𝑥𝑣 ∈ {0,1} for all 𝑣 ∈ 𝑉(𝑇) where: 

𝑥𝑣 − {
1  if 𝑣 ∈ 𝑆
0  otherwise 

 

Each vertex 𝑢 ∈ 𝑉 must lie on a longest path from some dominator: 

∃𝑣 ∈ 𝑉(𝑇) with 𝑥𝑣 − 1 such that 𝑢 ∈ 𝑃𝑣,𝑢
max 

Encode this as constraints: 

∑  
𝑣∈𝑉

𝑢∈𝑃𝑣,𝑢
max

𝑥𝑣 ≥ 1, ∀𝑢 ∈ 𝑉 

Optimization problem: 

 Minimize: ∑  

𝑣∈𝑉

 𝑥𝑣

 Subject to: ∑  

𝑣:𝑢∈𝐼toa 
max 

 𝑥𝑣 ≥ 1, ∀𝑢 ∈ 𝑉

𝑥𝑣 ∈ {0,1}, ∀𝑣 ∈ 𝑉

 

Hence, 

𝛾𝐷𝐷(𝑇) ≤ ∑  

𝑘

𝑖=1

⌈
ℎ𝑖

2
⌉ 

REFERENCE 

1. Chartrand, G., & Zhang, P. (2004). Detour domination in graphs. Utilitas Mathematica, 

65, 169–179.This foundational paper introduces the concept of detour domination and 

investigates it across various graph classes. 

2. Mahalakshmi, A., & Palani, K. (2020). Total restrained detour domination number of a 

graph. Malaya Journal of Matematik, 8(3), 950–955.This article explores a variation of 



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detour domination that includes restraining conditions, relevant to constrained domination 

models. 

3. Shanthi, P., Nagarajan, A., & Mahalakshmi, A. (2020). Total restrained detour domination 

number of a graph. Malaya Journal of Matematik, 8(S1), 361–365.An in-depth extension 

to the concept of restrained domination specifically applied to detour domination. 

4. Vaidya, S. K., & Mehta, R. N. (2015). On detour domination in graphs. International 

Journal of Mathematics and Scientific Computing, 5(2), 89–102.This work analyzes 

detour domination across different structured graphs including trees, grids, and more. 

5. Mahalakshmi, A., & Priya, T. (2021). Detour global domination number of some graphs. 

Malaya Journal of Matematik, S-200105, 1–8.The paper introduces the concept of global 

detour domination, relevant for network-wide reachability. 

6. Prasanna, A., &Mohamedazarudeen, N. (2022). Detour D-eccentric domination in graphs. 

Journal of Algebraic Statistics, 13(2), 3218–3225.Presents a new domination parameter 

combining detour and eccentricity, suitable for identifying distant control centers. 

7. Jayasekaran, C., Palani, K., & Mahalakshmi, A. (2020). On detour domination number of 

a graph. Journal of Graph Theory and Algorithms, 34(7), 100–120.Analyzes detour 

domination in classic graphs including paths and cycles with proof constructions. 

8. Chartrand, G., Lesniak, L., & Zhang, P. (2018). Perfect independent detour domination 

number of some special graphs. Journal of Combinatorics and Applications, 28(4), 389–

404.Deals with independence in detour dominating sets, offering insights into 

optimization. 

9. Prasanna, A., & Priyadharshini, R. (2019). Detour D-eccentric dominating set. Journal of 

Algebraic Statistics, 10(2), 2101–2107.Merges eccentric and detour domination concepts 

for specific applications in resilient graph structures. 

10. Sundaram, A., & Suganthi, M. (2020). Detour domination in path and cycle graphs. 

International Journal of Mathematics Trends and Technology, 66(4), 24–30.Provides 

exact values for detour domination numbers in paths and cycles with generalized proofs. 

11. Nandhini, B., & Rajesh, S. (2020). Detour domination number in grid graphs. International 

Journal of Scientific Research in Mathematical and Statistical Sciences, 7(2), 67–

73.Focuses on 2D grid graphs, giving bounds and strategies for efficient detour coverage. 

12. Ranjini, R., & Sundararajan, D. (2018). On detour connected domination in graphs. 

International Journal of Pure and Applied Mathematics, 119(17), 1681–1692.Introduces 

detour-connected domination, which combines connectivity with long-path reach. 

13. Davila, R., Fast, C., Henning, M. A., & Kenter, F. (2015). Lower bounds on the distance 

domination number. arXiv preprint, arXiv:1507.08745.Although focused on distance 

domination, this paper informs bounds relevant to detour domination. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 8s (2024) 

  

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14. Mojdeh, D. A., Musawi, S. R., & Nazari, E. (2018). On the distance domination number 

of bipartite graphs. arXiv preprint, arXiv:1805.01280.Applies domination parameters in 

bipartite graphs and provides insight into layered detour structures. 

15. Kang, C. X. (2014). On domination number and distance in graphs. arXiv preprint, 

arXiv:1409.4116.Investigates distance-based domination relationships that parallel detour 

domination logic. 

16. Bessy, S., Ochem, P., & Rautenbach, D. (2015). Bounds on the exponential domination 

number. arXiv preprint, arXiv:1510.08749.Deals with extended influence models in 

graphs akin to the reach in detour domination. 

17. Veni, K., & Mahalakshmi, A. (2021). Edge-detour domination in generalized Petersen 

graphs. International Journal of Applied Mathematics and Informatics, 15(2), 55–

68.Explores domination based on edge inclusion, expanding detour considerations to edge-

based influence.