Ratio Mathematica                                                                            Volume 44, 2022 

 
 

 
 

 

The Forcing Geodetic Cototal Domination 

Number of a Graph 

 

S. L. Sumi1 

V. Mary Gleeta2 

J. Befija Minnie3 

 
 

Abstract 

Let 𝑆 be a geodetic cototal domination set of 𝐺. A subset 𝑇 ⊆  𝑆 is called a forcing 
subset for 𝑆 if 𝑆 is the unique minimum geodetic cototal domination set containing 𝑇. 
The minimum cardinality T is the forcing geodetic cototal domination number of S is 

denotedby 𝑓𝛾𝑔𝑐𝑡 (𝑆), is the cardinality of a minimum forcing subset of S. The forcing 

geodetic cototal domination number of 𝐺,denoted by 𝑓𝛾𝑔𝑐𝑡 (𝑆), is 𝑓𝛾𝑔𝑐𝑡 (𝐺) =

 𝑚𝑖𝑛{𝑓𝛾𝑔𝑐𝑡 (𝑆)}, where the minimum is takenover all 𝛾𝑔𝑐𝑡-sets 𝑆 in 𝐺. Some general 

properties satisfied by this concept arestudied. It is shown that for every pair 𝑎, 𝑏 of 
integers with 0 ≤  𝑎 <  𝑏, 𝑏 ≥  2,there exists a connected graph 𝐺 such that 
𝑓𝛾𝑔𝑐𝑡 (𝐺)  =  𝑎 and 𝛾𝑔𝑐𝑡 (𝐺)  =  𝑏. where𝛾𝑔𝑐𝑡 (𝐺) isthe geodetic cototal dominating 

number of 𝐺. 
 

Keywords: geodetic set, cototal dominating set, geodetic cototal dominating set, 

geodetic cototal domination number, forcing geodetic cototal domination number. 

 

AMS Subject Classification: 05C12, 05C694 

 

 
1Research Scholar, Register No.20123042092007, Department of Mathematics, Holy Cross College 

(Autonomous), Nagercoil - 629004, Affiliated by Manonmaniam Sundaranar University, Abishekapatti, 

Tirunelveli - 627 012, Tamil Nadu, India. sumikrish123@gmail.com. 
2Assistant Professor, Department of Mathematics, T.D.M.N.S College, T.     Kallikulam - 627 113, 

Affiliated by Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, 

India. gleetass@gmail.com. 
3Assistant Professor, Department of Mathematics, Holy Cross College (Autonomous), Nagercoil - 

629004, Affiliated by Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil 

Nadu, India, Mail Id: befija@gmail.com 
4Received on June 11th, 2022.Accepted on Sep 9st, 2022.Published on Nov 30th, 2022.doi: 

10.23755/rm.v44i0.895. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY licence agreement 

 

93



S. L. Sumi, V. Mary Gleeta and J. Befija Minnie 

 

1. Introduction 

By a graph 𝐺 =  (𝑉, 𝐸), we mean a finite, undirected connected graph without 
loops or multiple edges. The order and size of 𝐺 are denoted by 𝑚and 𝑛respectively. 
For basic definitions and terminologies, we refer to [1,2]. For vertices 𝑢 and 𝑣 in a 
connected graph 𝐺, the distance𝑑(𝑢, 𝑣) is the length of a shortest 𝑢– 𝑣 path in 𝐺. A 𝑢– 𝑣 
path of length 𝑑(𝑢, 𝑣) is called a 𝑢– 𝑣geodesic. The eccentricity𝑒(𝑣) of a vertex 𝑣 in 𝐺 
is the maximum distance from 𝑣 and a vertex of 𝐺. The minimum eccentricity among 
the vertices of 𝐺 is the radius, 𝑟𝑎𝑑𝐺 or𝑟(𝐺) and the maximum eccentricity is its 
diameter, 𝑑𝑖𝑎𝑚𝐺of 𝐺. Let 𝑥, 𝑦 ∈ 𝑉and  let𝐼[𝑥, 𝑦] be the set of all vertices that lies in 
𝑥 − 𝑦 geodesic including 𝑥and 𝑦. Let 𝑆 ⊆ 𝑉(𝐺)and 𝐼[𝑆] = ⋃ 𝐼[𝑥, 𝑦]𝑥,𝑦∈𝑆 . Then 𝑆 is 

said to be a geodetic set of 𝐺, if 𝐼[𝑆] = 𝑉. The geodetic number𝑔(𝐺) of 𝐺is the 
minimum order of its geodetic sets and any geodetic set of order 𝑔(𝐺) is called a 𝑔-
setof 𝐺. A set 𝑆 ⊆ 𝑉 (𝐺) is called a dominating set if every vertex in 𝑉(𝐺)  − 𝑆 is 
adjacent to at least one vertex of 𝑆. The domination number, 𝛾(𝐺), of a graph 𝐺 denotes 
the minimum cardinality of such dominating sets of  𝐺. A minimum dominating set of a 
graph 𝐺 is hence often called as a 𝛾-set of 𝐺. The domination concept was studied in 
[3]. A dominating set 𝑆 of 𝐺 is a cototal dominating set if every vertex 𝑣 ∈ 𝑉 ∖ 𝑆 is not 
an isolated vertex in the induced subgraph  〈𝑉 ∖ 𝑆〉. The cototal domination number 
𝛾𝑐𝑡 (𝐺) of 𝐺 is the minimum cardinality of a cototal dominating set. The cototal 
domination number of a graph was studied in [4].   A set 𝑆 ⊆ 𝑉 is said to be a geodetic 
cototal dominating set of 𝐺, If𝑆 is both geodetic set and cototal dominating set of 𝐺. 
The geodetic cototal domination number of 𝐺 is the minimum cardinality among all 
geodetic cototal dominatingsets in 𝐺 and denoted by 𝛾𝑔𝑐𝑡 (𝐺). A geodetic cototal 

dominating set of minimumcardinality is called the 𝛾𝑔𝑐𝑡-set of 𝐺. The geodetic cototal 

domination number of agraph was studied in [6]. The following theorems are used in 

the sequel. 

 

Theorem 1.1. [6] Every end vertex of G belongs to every geodetic cototal dominating 

set of G. 

 

2. The forcing geodetic Cototal domination number of 

a graph 

Even though every connected graph contains a minimum geodetic cototal dominating 

sets, some connected graph may contain several minimum geodetic cototal dominating 

sets. For each minimum geodetic cototal dominating set S in a connected graph there is 

always some subset T of S that uniquely determines S as the minimum geodetic cototal 

dominating set containing T such “forcing subsets” are considered in this section.  The 

forcing concept was studied in [5] 

 

Definition 2.1. Let 𝑆 be a geodetic cototal domination set of 𝐺. A subset T ⊆ S is called 
a forcing subset for S if S is the unique minimum geodetic cototal domination set 

containing T. The minimum cardinality T is the forcing geodetic cototal domination 

94



The Forcing Geodetic Cototal Domination Number of a Graph 

 
 

number of S is denoted by 𝑓𝛾𝑔𝑐𝑡 (𝑆), is the cardinality of a minimum forcing subset of S. 

The forcing geodetic cototal domination number of 𝐺, denoted by𝑓𝛾𝑔𝑐𝑡 (𝑆),  is 

𝑓𝛾𝑔𝑐𝑡 (𝐺)  = min {𝑓𝛾𝑔𝑐𝑡 (𝑆)}, where the minimum is taken over all 𝛾𝑔𝑐𝑡-setsS in G. 

 

Example 2.2. For the graph 𝐺 of Figure 2.1, 𝑆1  =  {𝑣3, 𝑣6, 𝑣7} and 𝑆2  =
 {𝑣2, 𝑣5, 𝑣7}are the only two 𝛾𝑔𝑐𝑡-sets of 𝐺 so that 𝛾𝑔𝑐𝑡 (𝐺) = 3 and 𝑓𝛾𝑔𝑐𝑡(𝑆1) = 

𝑓𝛾𝑔𝑐𝑡 (𝑆2)  =  1 sothat 𝑓𝛾𝑔𝑐𝑡 (𝐺)  =  1. 

 

 
The following result follows immediately from the definitions of the geodetic cototal 

domination number and the forcing geodetic cototal domination number of a connected 

graph 𝐺. 
 

Theorem 2.3. For every connected graph 𝐺, 0 ≤ 𝑓𝛾𝑔𝑐𝑡 (𝐺)  ≤ 𝛾𝑔𝑐𝑡 (𝐺). 

 

Remark 2.4. The bounds in Theorem 2.3 are sharp. For the complete graph 𝐺 =
 𝐾𝑛,𝑆 =  𝑉 is the unique 𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺)  = 0. Also, the bounds in 

Theorem2.3 can be strict. For the graph 𝐺 given in Figure 2.1, 𝛾𝑔𝑐𝑡 (𝐺)= 3 and 

𝑓𝛾𝑔𝑐𝑡 (𝐺)  = 1. Thus 0 < 𝑓𝛾𝑔𝑐𝑡 (𝐺)  < 𝛾𝑔𝑐𝑡 (𝐺). 

 

Theorem 2.5. Let G be a connected graph. Then 

(a) 𝑓𝛾𝑔𝑐𝑡 (𝐺)  = 0 if and only if G has a unique minimum 𝛾𝑔𝑐𝑡-set. 

(b) 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 1 if and only if G has at least two minimum 𝛾𝑔𝑐𝑡-sets, one of which isa  

unique minimum 𝛾𝑔𝑐𝑡-set containing one of its elements and 

(c) 𝑓𝛾𝑔𝑐𝑡 (𝐺)  = 𝛾𝑔𝑐𝑡 (𝐺) if and only if no 𝛾𝑔𝑐𝑡-set of 𝐺 is the unique minimum 𝛾𝑔𝑐𝑡-set 

containing any of its proper subsets. 

 

𝑣1 

𝑣6 𝑣2 

𝑣5 

𝑣3 

𝐺  
Figure 2.1 

 

𝑣4 

𝑣7 

95



S. L. Sumi, V. Mary Gleeta and J. Befija Minnie 

 

Definition 2.6. A vertex 𝑣 of a connected graph 𝐺 is said to be a geodetic cototal 
dominating vertex of 𝐺 if 𝑣 belongs to every 𝛾𝑔𝑐𝑡-set of 𝐺. 

 

Example 2.7. For the graph 𝐺 given in Figure 2.2, 𝑆1 = {𝑣1, 𝑣3, 𝑣6} and 𝑆2  =
{𝑣1, 𝑣3, 𝑣5} are the only two minimum 𝛾𝑔𝑐𝑡-sets of 𝐺 so that {𝑣1, 𝑣3} is the geodeticco-

total dominating vertex of G.Then 𝑓𝛾𝑔𝑐𝑡 (𝐺)  ≤ 𝛾𝑔𝑐𝑡 (𝐺)–  | 𝑊 |. 

 

 
 

Remark 2.9. The bound in Corollary 2.7 is sharp. For the graph 𝐺 of Figure 2.2, 𝑆1 =
 {𝑣1, 𝑣3, 𝑣6} and 𝑆2 =  {𝑣1, 𝑣3, 𝑣5} are the only two minimum 𝛾𝑔𝑐𝑡-sets of 𝐺 so that 

𝑓𝛾𝑔𝑐𝑡 (𝑆1)  =  𝑓𝛾𝑔𝑐𝑡  (𝑆2)  =  1 so that 𝛾𝑔𝑐𝑡 (𝐺)  =  3 and 𝑓𝛾𝑔𝑐𝑡 (𝐺)   =  1. Also, 𝑊 =

 {𝑣1, 𝑣3} is theset of all geodetic cototal dominating vertices of 𝐺. Now,𝛾𝑔𝑐𝑡 (𝐺) − |𝑊| = 

3 − 2 = 1. Thus 𝑓𝛾𝑔𝑐𝑡 (𝐺)  < 𝛾𝑔𝑐𝑡 (𝐺)  −  |𝑊|. Also, the bounds in Theorem 2.7 can be 

strict. 

 

Theorem 2.10. For the complete bipartite graph 𝐺 =  𝐾𝑟,𝑠 (1 ≤ 𝑟 ≤ 𝑠), 

𝑓𝛾𝑔𝑐𝑡 (𝐺)  =  {
0,   𝑖𝑓 1 ≤ 𝑟 ≤ 3
4, 𝑖𝑓4 ≤  r ≤  s

 

Proof: Let 𝑈 =  {𝑢1, 𝑢2, . . . , 𝑢𝑟 } and 𝑊 =  {𝑤1, 𝑤2, . . . , 𝑤𝑠} be the bipartite sets of 𝐺. 
For 1 ≤ 𝑟 ≤  3. Let 𝑆 =  𝑈 ∪ 𝑊 is the unique 𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. 

Let1 ≤ 𝑟 ≤  3. If 𝑟 ≥  4, then every 𝛾𝑔𝑐𝑡 (𝐺)-set is of the form 𝑆 =

 {𝑢𝑖1 , 𝑢𝑖2 , 𝑤𝑗1 , 𝑤𝑗2 }where1≤ 𝑖1 ≤ 𝑖2 ≤ 𝑟 and 1 ≤ 𝑗1 ≤ 𝑗2 ≤ 𝑠. Since 𝑆 is not the unique 

geodetic cototaldominating set containing any of its proper subset, By Theorem 

𝑓𝛾𝑔𝑐𝑡 (𝐺) = 4.∎ 

 

Theorem 2.11. For the wheel 𝐺 =  𝐾𝑛 +  𝐶𝑛−1 (𝑛 ≥  5), 

𝑓𝛾𝑔𝑐𝑡 (𝐺) ={
1, 𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛
2,   𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑

 

𝑣1 

𝑣2 

𝑣5 𝑣6 

𝑣3 

𝑣4 𝐺  
Figure 2.2 

 

FigurFigFure 2.1 

Figure 2.1 

e 1.1 

96



The Forcing Geodetic Cototal Domination Number of a Graph 

 
 

Proof: Let 𝑥 be the central vertex of 𝐺 and 𝐶𝑛−1 be 𝑣1, 𝑣2, … , 𝑣𝑛−1, 𝑣𝑛 . 
Case 1: 𝑛 is even. 
Then 𝑆1 =  {𝑣1, 𝑣3, 𝑣5, . . . , 𝑣𝑛−3, 𝑣𝑛−1}, 𝑆2 =  {𝑣2, 𝑣4, 𝑣6, . . . , 𝑣𝑛−2, 𝑣𝑛 } are the only 
two𝛾𝑔𝑐𝑡-sets of 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝑆1)  = 𝑓𝛾𝑔𝑐𝑡 (𝑆2)  =  1 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺)  =  1. 

Case 2: 𝑛 is odd. 
Then 𝑆1  =  {𝑣1, 𝑣3, 𝑣5, … , 𝑣𝑛 }, 𝑆2 =  {𝑣2, 𝑣4, 𝑣6, … , 𝑣𝑛−1, 𝑣1}, … , 𝑆𝑛 2⁄

=  {𝑣𝑛
2⁄

, 𝑣𝑛
2⁄ +1

 , 

 . . . , 𝑣1, 𝑣3, 𝑣𝑛 2⁄ −1
}are the 𝑛/2 𝛾𝑔𝑐𝑡-sets of 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝑆1)  =  𝑓𝛾𝑔𝑐𝑡 ( 𝑆2)  = . . . =

 4𝑓𝛾𝑔𝑐𝑡 (𝑆𝑛 2⁄
)  =  2 sothat 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 2.∎ 

 

Theorem 2.12. For the helm graph 𝐺 =  𝐻𝑟 , 𝐺 =  𝑇, 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0, for 𝑛 ≥  6. 

Proof: Let 𝑆 be the set of end vertices and the cut vertices of 𝐺. Then 𝑆 is theunique 
𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0.∎ 

 

Theorem 2.13. For the Triangular snake graph 𝐺 =  𝑇𝑟, 𝑓𝛾𝑔𝑐𝑡 (𝐺)  = 0. 

Proof: Let 𝑆 be the set of extreme vertices of 𝐺. Then S is the unique 𝛾𝑔𝑐𝑡-set of 𝐺so 

that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0.∎ 

 

Theorem 2.14. For the fan graph 𝐹𝑛  = 𝐾1  +  𝑃𝑛−1, 

𝑓𝛾𝑔𝑐𝑡 (𝐺) = {
0, 𝑖𝑓𝑛 − 1 𝑖𝑠𝑜𝑑𝑑

1,     𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛
 

Proof: Let 𝑉 (𝐾1)  =  {𝑥} and 𝑉(𝑃𝑛−1)  = {𝑣1, 𝑣2, . . . , 𝑣𝑛−1}. 
Let 𝑛 −  1 is odd. Let 𝑛 −  1 =  2𝑘 +  1. Then 𝑆 =  {𝑣1, 𝑣3, 𝑣5, . . . , 𝑣2𝑘+1} is the 
unique𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. 

Let 𝑛 −  1 be even. Let 𝑛 −  1 =  2𝑘. Then 𝑆1 =  {𝑣1, 𝑣3, 𝑣5, . . . , 𝑣2𝑘−1, 𝑣2𝑘 }, 𝑆2  = 
{𝑣1, 𝑣3, 𝑣5, . . . , 𝑣2𝑘−2 , 𝑣2𝑘 , 𝑣2}are the two 𝛾𝑔𝑐𝑡-sets of 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝑆1)  =

𝑓𝛾𝑔𝑐𝑡 (𝑆2) = 1.so that 𝑓𝛾𝑔𝑐𝑡 (𝐺)  =  1. ∎ 

 

Theorem 2.15. For the Banana tree graph 𝐺 =  𝐵𝑟,𝑠, 𝑓𝛾𝑔𝑐𝑡 (𝐺)   =  0. 

Proof: Let 𝑥 be the centre vertex of 𝐺 and the set of all end vertices of 𝐺. Then𝑆 =
 𝑍 ∪  {𝑥} is the unique 𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0.∎ 

 

Theorem 2.16. For the sunflower graph 𝐺 =  𝑆𝐹𝑛, 𝑓𝛾𝑔𝑐𝑡 (𝐺) =  0. 

Proof: Let 𝑆 be the set of extreme vertices of 𝐺. Then 𝑆 is the unique 𝛾𝑔𝑐𝑡-set of 𝐺. So 

that 𝑓𝛾𝑔𝑐𝑡 (𝐺) =  0. ∎ 

 

Theorem 2.17. For every pair𝑎, 𝑏of integers with 0 ≤ 𝑎 < 𝑏, 𝑏 ≥  2, there exists a 
connected graph 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝐺)  =  𝑎 and 𝛾𝑔𝑐𝑡 (𝐺) =  𝑏. 

Proof: Let 𝑃 ∶  𝑢, 𝑣, 𝑧be a path of order three. Let 𝑃𝑖 : 𝑢𝑖 , 𝑣𝑖  (1 ≤ 𝑖 ≤ 𝑎) be a copyof 
path on two vertices. Let 𝐻be a graph obtained from 𝑃and 𝑃𝑖 (1 ≤ 𝑖 ≤ 𝑎) byjoining 
each 𝑢𝑖  (1 ≤ 𝑖 ≤ 𝑎) with 𝑣 and each 𝑣𝑖  (1 ≤ 𝑖 ≤ 𝑎) with 𝑧. Let 𝐺 be thegraph 
obtained from 𝐻 by introducing new vertices𝑧1, 𝑧2, . . . , 𝑧𝑏−𝑎+1 joining each𝑧𝑖  (1 ≤ 𝑖 ≤
𝑎) with 𝑧. The graph 𝐺 is given in Figure 2.4. 

97



S. L. Sumi, V. Mary Gleeta and J. Befija Minnie 

 

First, we show that γgct(G) =  b. Let Z =  {u, z1, z2, . . . , zb−a+1} be the set of 

endvertices of G. By Theorem 1.1, Z is a subset of every geodetic cototal dominating set 
of G. Let Hi = {ui, vi}. Then it is easily observed that every geodetic cototaldominating 
set containing at least one vertex from each Hi(1 ≤ i ≤ a) and soγgct(G)≥ b– a +  a =

 b. Let S =  Z ∪  {u1, u2, . . . , ua}. Then S is a minimum geodeticcototal dominating set 
of G so that γgct(G) = b. 

Next, we prove that fγgct(G)  =  a. Since every geodetic co-total dominating set 

ofGcontains Z, it follows thatfγgct(G)  ≤ γgct(G) −  | Z | =  b − (b − a)  =  a.Now, 

since γgct(G) = b and every γgct-set of G contains Z, it is easily seen that everyγgct-set 

of G is of the form S =  Z ∪ {c1, c2, . . . , ca}, where ci ∈ Hi (1 ≤ i ≤ a). Let T beany 
proper subset of S with | T | < a. Then there exists an edge ej (1 ≤ j ≤ a) suchthat ej ∉

T. Let fj be an edge of Hj distinct from ej. Then W1 =  (S −  {ej}  ∪  {fj}is a γgct-set 

properly containing T. Thus W is not the unique γgct-set containingT. Thus T is not a 

forcing subset of S. This is true for all minimum geodetic cototaldominating sets of G 
and so it follows that fγgct(G) =  a. 

 

 
 

3. Conclusion 
In this paper we studied the concept of forcing geodetic cototal domination number of 

some standard graphs some general properties satisfied by this concept are studied. In 

future studies, the same concept is applied for the other graph operations. 

 

𝑧𝑏−𝑎−1 

𝑧2 

𝑧1 

  𝑣2 

𝑢1 𝑢 

𝐺 

Figure 2.3 

 

𝑣 

 

 

𝑧 𝑣1 

𝑢2 

𝑢3 

𝑢4 

𝑢𝑎 

𝑣4 

𝑣3 

𝑣𝑎  

98



The Forcing Geodetic Cototal Domination Number of a Graph 

 
 

References 
 

[1] F. Harary, Graph Theory, Addison – Wesley, (1969). 

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[3] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in 

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of a graph, Journal of Discrete Mathematical Sciences and Cryptography, 2 (2), (1999), 

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[5] Gary Chartrand and P. Zhang, The Forcing Geodetic number of a Graph, 

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