Untitled


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

894 https://internationalpubls.com 

 

1T. Sindhuja, 2*V. Maheswari and 3V. Balaji 

1Research Scholar, Department of Mathematics, Vels Institute of Science Technology and Advanced Studies, Chennai, 

Tamil Nadu, India.  

Email: sindhutg86@gmail.com 

2,*Research Supervisor, Professor, Department of Mathematics, Vels Institute of Science Technology and Advanced 

Studies, Chennai, Tamil Nadu, India.  

Corresponding author:

3Associate Professor, Department of Mathematics, Sacred Heart College, Tirupattur, Tamil Nadu, India. 

Email: pulibala70@gmail.com 

 

Article History: 

Received: 12-11-2024 

Revised: 17-12-2024 

Accepted: 06-01-2025 

Abstract:  

Let D be a subset of V and  𝑢 ∈ 𝐷  the out degree of u is defined by |𝑁(𝑢) ∩ (𝑉 − 𝐷)| and 

denoted by 𝑜𝑑𝐷(𝑢). A dominating set D of V is said to be co- total dominating set if the sub 

graph induced  by 𝑉 − 𝐷 has no isolated vertices Based on the concepts we introduce a new 

domination called co-total 2 - ODED number. Here the proposed domination number verified 

for some general and some named graphs. 

Keywords: Two out degree, domination number, isolated vertices, co-total  

1. 

The graphs considered here are simple and without isolated vertices defined from [1-5]. In a graph 

G=(V,E),  |V|=n and |E|=m.  Let 𝑢 be vertex and N(u) denotes the open neighborhood and defined by 𝑁(𝑢) = {𝑣 ∈ 𝑉(𝐺)/ 𝑢𝑣 ∈ 𝐸(𝐺)} The degree of a vertex u is denoted 𝑑𝑒𝑔(𝑢) and defined by 𝑑𝑒𝑔(𝑢) = |𝑁(𝑢)|. If the vertex u of a graph is isolated vertex if  𝑑𝑒𝑔(𝑢) = 0.   
 O. Ore [10] and C. Berge [4] introduced the concept of domination number. A sub set D of V 

is said to dominating set if each vertex of V is adjacent to some vertex in D. The domination number 𝛾(𝐺) of G is the minimum cardinality of a dominating set. Kulli V.R., Jankiram B and Radha R Iyer 

[6] introduce co-total domination number 𝛾𝑐𝑡(𝐺).  

dominating set if the sub graph induced  by V-D has no isolated vertices. The minimum number of 

vertices of co-total dominating set is called co-total domination number and it is represented by 𝛾𝑐𝑡(𝐺). 

Sahal. A and V. Mathad [11] introduce the 2- ODED number. A dominating set D  of V(G) is said to 

be 2- ODED set for any vertices 𝑢, 𝑣 ∈ 𝐷 such that |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2 where 𝑜𝑑𝐷(𝑢) =|𝑁(𝑢) ∩ (𝑉 − 𝐷)|. The minimum represented of vertices in 2- ODED set is called 2- ODED number 

and it is denoted by 𝛾2𝑜𝑒(𝐺). Based on above domination number, here we introduce a new domination 

parameter  called co - total 2- ODED (𝛾𝑐𝑡2𝑜𝑒(𝐺) − 𝑠𝑒𝑡) number.  

2. Co-Total 2 - ODED Number 

Definition 2.1 A sub set D  

vertices 𝑢, 𝑣 ∈ 𝐷 such that |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2 where 𝑜𝑑𝐷(𝑢) = |𝑁(𝑢) ∩ (𝑉 − 𝐷)| and the 

maheswari.sbs@vistas.ac.in

Co - Total 2 - ODED Number in Graphs

Introduction

A dominating set D of V is said to be co- total

of V is called co- total 2 - ODED set if D is dominating set and for any

mailto:sindhutg86@gmail.com
mailto:sasikala.sbs@velsuniv.ac.in
mailto:pulibala70@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

895 https://internationalpubls.com 

induced sub graph < 𝑉 − 𝐷 >has no vertices of degree zero. The minimum cardinality of a co-total 2 

- ODED  set is called co-total 2 - ODED number and it is represented  by 𝛾𝑐𝑡2𝑜𝑒(G). 

Example 2.2  

 

Figure 1. CO-total 2 - ODED number 

  

Take D= {4,8,9,10} 𝑎𝑛𝑑 𝑉 − 𝐷 = {1,2,3,5,6,7} 𝑜𝑑𝑆(4) = |𝑁(4) ∩ (𝑉 − 𝐷)| 
               = |[1,2,9} ∩ {1,2,3,5,6,7}| = |{1,2}| = 2 𝑜𝑑𝑆(8) = |𝑁(8) ∩ (𝑉 − 𝐷)| 
               = |[3,7,9} ∩ {1,2,3,5,6,7}| = |{3,7}| = 2 𝑜𝑑𝑆(9) = |𝑁(4) ∩ (𝑉 − 𝐷)| 
               = |[4,8,10} ∩ {1,2,3,5,6,7}| = |{∅}| = 0 𝑜𝑑𝑆(10) = |𝑁(10) ∩ (𝑉 − 𝐷)| 
               = |[5,6,9} ∩ {1,2,3,5,6,7}| = |{5,6}| = 2 

        Then any u,v∈ 𝐷, |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2 

So D= {4,8,9,10}form 2- ODED set of  𝐺 and the sub graph induced  by <𝑉 − 𝐷> has no vertices of 

degree zero. Hence D is a co-total 2- ODED set with minimum cardinality.Hence 𝛾𝑐𝑡2𝑜𝑒(𝐺) = 3 

Observation 2.3 A graph G with p vertices, then  2≤ 𝛾𝑐𝑡2𝑜𝑒(𝐺) ≤ 𝑛. 

Observation 2.4 

1. For any complete graph:𝛾𝑐𝑡2𝑜𝑒(𝐾𝑛) = 2 

2. For any  star 𝐾1,𝑛, then 𝛾𝑐𝑡2𝑜𝑒(𝐾1,𝑛−1) = 𝑛 

3. For any complete bipartite graph 𝐾𝑝,𝑞 , is 𝛾𝑐𝑡2𝑜𝑒(𝐾𝑝,𝑞) = { 2             𝑖𝑓 |𝑝 − 𝑞| ≤ 2𝑝 + 𝑞 − 2    𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒                         
4. For any cycle 𝐶𝑛, then 𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛) = 𝑛 − 2 

5. For the Path  𝑃𝑛,  𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛) =𝑛 − 2, n≥ 2 

6. For the double star 𝑆𝑟,𝑡, 𝛾𝑐𝑡2𝑜𝑒(𝑆𝑟,𝑡)= 𝑟 + 𝑡 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

896 https://internationalpubls.com 

3. Co-total 2 - ODE D number for different graphs 

Here begin the investigation of the co-total 2- ODED number by computing its values for some general 

of graphs.  

Theorem: 3.1 For any  combo graph  𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛+) = 𝑛 

Proof:  Let V(𝑃𝑛+) = {𝑣1, 𝑣2𝑣3 … . 𝑣𝑛, 𝑣𝑛+1, 𝑣𝑛+2, 𝑣𝑛+3, … . 𝑣2𝑛}. Here {𝑣1, 𝑣2𝑣3 … . 𝑣𝑛} be the 

vertices of the path { 𝑣𝑛+1, 𝑣𝑛+2, 𝑣𝑛+3, … . 𝑣2𝑛} be the vertices of degree one is  incident with  each 

vertex of the path. Let D= {𝑣𝑛+1, 𝑣𝑛+2, 𝑣𝑛+3, … . 𝑣2𝑛} be minimal dominating set and 𝑉 − 𝐷= 

{𝑣1, 𝑣2𝑣3 … . 𝑣𝑛}. Each vertices D has a neighborhood 𝑉 − 𝐷 and 𝑜𝑑𝐷(𝑣𝑛+𝑖) = |𝑁(𝑣𝑛+𝑖)⋂(𝑉 −𝐷)|=1 for all i=1,2,3……n. then |𝑜𝑑𝐷(𝑣𝑛+𝑖) − 𝑜𝑑𝐷(𝑣𝑛+𝑗)| ≤ 2. Then D is 2- ODED set  and < 𝑉 −𝐷 > has no vertices of degree zero. Then D is minimal co-total 2- ODED set . Then 𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛+) =|𝐷| = 𝑛. 

Theorem 3.2 For any crown graph 𝐶𝑛+, 𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛+) = 𝑛 

Proof: Let V(𝐶𝑛+) = {𝑣1, 𝑣2, 𝑣3 … . 𝑣𝑛, 𝑣𝑛+1, 𝑣𝑛+2 … . 𝑣2𝑛} . Here {𝑣1, 𝑣2, 𝑣3 … . 𝑣𝑝} be the vertices of 

cycle 𝐶𝑝 𝑎𝑛𝑑 {𝑣𝑛+1, 𝑣𝑛+2, … . , 𝑣2𝑛}  be the vertices of degree zero which adjacent to 𝑣1, 𝑣2, 𝑣3 … . 𝑣𝑛 

respectively. Let us take D= {𝑣𝑛+1, 𝑣𝑛+2, … . , 𝑣2𝑛} be dominating set with minimal  cardinality and 𝑉 − 𝐷 = {𝑣1, 𝑣2, 𝑣3 … . 𝑣𝑛, }  then 𝑜𝑑𝐷(𝑣𝑛+𝑖) = |𝑁(𝑣𝑛+𝑖) ∩ (𝑉 − 𝐷)   = |{𝑣𝑖} ∩ {𝑣1, 𝑣2, 𝑣3 … . 𝑣𝑛}|   
= |{𝑣𝑖}|=1. Then |𝑜𝑑𝐷(𝑣𝑖) − 𝑜𝑑𝐷(𝑣𝑗)| = 0 < 2. 

Hence  D is a 2 - ODED set and the sub graph induced by  𝑉 − 𝐷 has no vertices of degree zero. Then 𝛾𝑐𝑡2𝑜𝑒(𝐶𝑛+) = |𝐷|= 𝑛. 

Theorem:3.3 For any triangular snake graph 𝛾𝑐𝑡2𝑜𝑒(𝑛𝐶3) = 𝑛 − 1 

Proof: The graph G contains 2𝑛 − 1 vertices and 𝑛 − 1 triangles. The upper vertices labeled from 𝑣1 

to 𝑣𝑛−1 and the lower vertices are labeled form 𝑣𝑛to 𝑣2𝑛−1. Let 𝐷 = {𝑣1, 𝑣2, … … . 𝑣𝑛−1}and V−𝐷 ={𝑣𝑛, 𝑣𝑛+1, 𝑣𝑛+2, 𝑣𝑛+3 … … . 𝑣2𝑛−2, 𝑣2𝑛−1}. 

The vertex 𝑣1 is adjacent to 𝑣𝑛, 𝑣𝑛+1. The vertex 𝑣2 is adjacent to 𝑣𝑛+1, 𝑣𝑛+2. The vertex 𝑣𝑛−1 is 

adjacent to 𝑣2𝑛−2, 𝑣2𝑛−1. Hence the set {𝑣𝑛+1, 𝑣𝑛+3 … … . . 𝑣2𝑛−2} is minimum dominating set. Now 𝑜𝑑𝐷(𝑣1) = |𝑁(𝑣1) ∩ (𝑉 − 𝐷)| = |{𝑣𝑛, 𝑣𝑛+1}| = 2 𝑜𝑑𝐷(𝑣2) = |𝑁(𝑣2) ∩ (𝑉 − 𝐷)| = |{𝑣𝑛+1, 𝑣𝑛+2}| = 2 and 𝑜𝑑𝐷(𝑣𝑛−1) = |𝑁(𝑣𝑛−1) ∩ (𝑉 − 𝐷)| =|{𝑣2𝑛−2, 𝑣2𝑛−1}| = 2 .  
Hence |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣 ∈ 𝐷, and D is minimum 2 - ODE dominating set  and 

Clearly < 𝑉 − 𝐷 >
set. Hence 𝛾𝑐𝑡2𝑜𝑒(𝐺) = 𝑛 − 1. 
Theorem 3.4 For any double triangular snake graph 𝛾𝑐𝑡2𝑜𝑒(𝐷(𝑛𝐶3)) = 𝑛 + 1 

Proof:  Let V(𝐷(𝑛𝐶3))= {𝑢1, 𝑢2, 𝑢3 … … 𝑢𝑛+1, 𝑣1, 𝑣2, 𝑣3 … … 𝑣𝑛,𝑤1, 𝑤2, 𝑤3 … … 𝑤𝑛}. Here 

{𝑢1, 𝑢2, 𝑢3 … … . . 𝑢𝑛+1} be the vertices of path 𝑃𝑛. From path  𝑃𝑛 join  𝑢𝑖 and 𝑢𝑖+1 to a new edges𝑣𝑖 

D is minimal co- total 2- ODED set . So

has no vertices of degree zero. Hence D is minimum co- total 2- ODE dominating



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

897 https://internationalpubls.com 

by  edges 𝑢𝑖𝑣𝑖  and 𝑢𝑖+1𝑣𝑖, for i=1,2,3… … ..n and join  𝑢𝑖 and 𝑢𝑖+1 to a new edges 𝑤𝑖 by  edges 𝑢𝑖𝑤𝑖 
and 𝑢𝑖+1𝑤𝑖 for i=1,2,3… … ..n 

Take the vertices of path 𝑃𝑛, D= {𝑢1, 𝑢2, 𝑢3 … … 𝑢𝑛+1} and 𝑉 − 𝐷 ={𝑣1, 𝑣2, 𝑣3 … … 𝑣𝑛, 𝑤1, 𝑤2, 𝑤3 … … 𝑤𝑛}. Clearly D is a dominating 𝑉 − 𝐷. Now  𝑜𝑑𝐷(𝑢𝑖) =|𝑁(𝑢𝑖) ∩ (𝑉 − 𝐷)| = |{𝑣𝑖, 𝑤𝑖}| = 2 for  𝑖 = 1, 𝑛 + 1 and 𝑜𝑑𝐷(𝑢𝑖) = |𝑁(𝑢𝑖) ∩ (𝑉 − 𝐷)| =|{𝑣𝑖−1, 𝑣𝑖 , 𝑤𝑖−1, 𝑤𝑖}| = 4 for 𝑖 = 2 … . . 𝑛. Hence|𝑜𝑑𝐷(𝑢𝑖) − 𝑜𝑑𝐷(𝑢𝑗)| ≤ 2 for all 𝑢𝑖, 𝑢𝑗  ∈ 𝐷, and D 

is minimum 2 - ODED set and  clearly < 𝑉 − 𝐷 > has no vertices of degree zero. So D is minimum 𝛾𝑐𝑡2𝑜𝑒(𝐷(𝑛𝐶3)) = 𝑛 + 1 

Definition 3.5 The square of a given graph G denoted by 𝐺2 has the same number vertices as of G and 

has a vertices are adjacent in 𝐺2 if they are at distance of one or two apart in G. 

Theorem 3.6 For any square of bistar graph 𝛾𝑐𝑡2𝑜𝑒(𝐵𝑝,𝑞2 ) = 𝑝 + 𝑞 

Proof: Consider  Bistar𝐵𝑝,𝑞, with vertices {𝑢, 𝑣, 𝑢1, 𝑢2, 𝑢3 … … 𝑢𝑝, 𝑣1, 𝑣2, 𝑣3 … … 𝑣𝑞} where 𝑢𝑖 , 𝑣𝑖are 

pendant vertices which is adjacent u and v respectively and  u and v adjacent.   

Take D= {𝑢1, 𝑢2, 𝑢3 … … 𝑢𝑝, 𝑣1, 𝑣2, 𝑣3 … … 𝑣𝑞} and 𝑉 − 𝐷 = {𝑢, 𝑣}. Clearly D is a dominating 𝑉 − 𝐷. 

Now  𝑜𝑑𝐷(𝑢𝑖) = |𝑁(𝑢𝑖) ∩ (𝑉 − 𝐷)| = |𝑉 − 𝐷| = 2 and 𝑜𝑑𝐷(𝑣𝑖) = |𝑁(𝑣𝑖) ∩ (𝑉 − 𝐷)| = |𝑉 − 𝐷| =2  . Hence|𝑜𝑑𝐷(𝑢𝑖) − 𝑜𝑑𝐷(𝑢𝑖)| = 0 ≤ 2  and D is minimum co-total 2 - ODED set. Clearly < 𝑉 −𝐷 > has no vertices of degree zero. So D is a minimum co- total 2 - ODED set. Hence 𝛾𝑐𝑡2𝑜𝑒(𝐵𝑝,𝑞2 ) =2. 

Theorem 3.7 For any path 𝑃𝑛, 𝛾𝑐𝑡2𝑜𝑒(𝑃𝑛2) = {2 ⌈𝑛7⌉ + 1  𝑖𝑓 𝑛 ≡ 0 𝑜𝑟 6 (𝑚𝑜𝑑 7)2 ⌈𝑛7⌉            𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒  

Proof: Let 𝑉(𝑃𝑛2) = {𝑣1, 𝑣2, 𝑣3, … . . , 𝑣𝑛} be the vertex set where deg(𝑣1)= deg(𝑣𝑛)=2, deg(𝑣2)= 

deg(𝑣𝑛−1)=3 and deg(𝑣𝑖)=4  for all i=1,2,3……n-2 

Case: 1 𝑛 ≡ 0 𝑜𝑟 6 (𝑚𝑜𝑑 7) 

If 𝑛 ≡ 0  (𝑚𝑜𝑑 7) we have 𝐷 = {(𝑣7𝑖+2, 𝑣7𝑖+4) ∪ {𝑣𝑛−1}} for 0 ≤ 𝑖 ≤ ⌈𝑛7⌉ − 1 and  

If 𝑛 ≡ 6  (𝑚𝑜𝑑 7) we have 𝐷 = {(𝑣7𝑖+2, 𝑣7𝑖+4) ∪ {𝑣𝑛}} for 0 ≤ 𝑖 ≤ ⌈𝑛7⌉ − 1 

Now 𝑜𝑑𝐷(𝑣7𝑖+2) = |𝑁(𝑣7𝑖+2) ∩ (𝑉 − 𝐷)| = 3 ,  𝑜𝑑𝐷(𝑣7𝑖+4) = |𝑁(𝑣7𝑖+4) ∩ (𝑉 − 𝐷)| = 4 , 𝑜𝑑𝐷(𝑣𝑛−1) = |𝑁(𝑣𝑛−1) ∩ (𝑉 − 𝐷)| = 2 and 𝑜𝑑𝐷(𝑣𝑛) =|𝑁(𝑣𝑛) ∩ (𝑉 − 𝐷)| = 2 

Hence |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2. For any 𝑢, 𝑣 ∈ 𝐷. 
Here < 𝑉 − 𝐷 >has no zero degree vertices then D is co-total 2 - ODED set also 𝐷 − {𝑢} is no zero 

degree vertices then D is a minimal co-total 2- ODED set.  𝛾𝑡2𝑜𝑒(𝑃𝑛2) = 2 ⌈𝑛7⌉ + 1   
Case: 2  𝑛 ≢ 0 𝑜𝑟 6 (𝑚𝑜𝑑 7) 

co- total 2- ODED set. Hence



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

898 https://internationalpubls.com 

If 𝑛 ≡ 1 𝑜𝑟 2 𝑜𝑟 3 𝑜𝑟 4 (𝑚𝑜𝑑 7) we have 𝐷 = {(𝑣7𝑖+2, 𝑣7𝑖+4) ∪ {𝑣𝑛−2,𝑣𝑛}} for 0 ≤ 𝑖 ≤ ⌊𝑛7⌋ − 1 and  

If 𝑛 ≡ 5  (𝑚𝑜𝑑 7) we have 𝐷 = {(𝑣7𝑖+2, 𝑣7𝑖+4)} for 0 ≤ 𝑖 ≤ ⌊𝑛7⌋ 
Now 𝑜𝑑𝐷(𝑣7𝑖+2) = |𝑁(𝑣7𝑖+2) ∩ (𝑉 − 𝐷)| = 3 ,  𝑜𝑑𝐷(𝑣7𝑖+4) = |𝑁(𝑣7𝑖+4) ∩ (𝑉 − 𝐷)| = 3 , 𝑜𝑑𝐷(𝑣𝑛) = |𝑁(𝑣𝑛) ∩ (𝑉 − 𝐷)| = 2 and 𝑜𝑑𝐷(𝑣𝑛−2) =|𝑁(𝑣𝑛−2) ∩ (𝑉 − 𝐷)| = 2 

Hence |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2. For any 𝑢, 𝑣 ∈ 𝐷. 
Here < 𝑉 − 𝐷 > has no vertices of degree zero then D is a co-total 2 - ODED set then  𝛾𝑡2𝑜𝑒(𝑃𝑛2) =2 ⌊𝑛7⌋. 
Definition 3.8 The semi total point graph T2(G) of G is the graph whose vertex set is 𝑉(𝐺) ∪ 𝐸(𝐺), 

whose vertices are adjacent if they adjacent vertices of G or one is a vertex of G and another edge of 

G incident with it.  

Theorem 3.9 For any cycle Cn, γct2oe T2(Cn)) = n − 1  forn ≥ 3 

Proof: Let vertex set of  V(C𝑛) = {𝑣1, 𝑣2, … , 𝑣𝑛} and edge set E(C𝑛) = {𝑒1, 𝑒2, … , 𝑒𝑛}. Now, V(T2(C𝑛)) 

= {𝑣1, 𝑣2, … , 𝑣𝑛, 𝑒1, 𝑒2, … , 𝑒𝑛} be vertices of T2(C𝑛). Let D = {𝑒1, 𝑒2, … , 𝑒𝑛−3, 𝑒𝑛, 𝑣𝑛−1} be the minimal 

dominating set of T2(C𝑛) then V−D = {𝑣1, 𝑣2, … , 𝑣𝑛−2, 𝑣𝑛, 𝑒𝑛−2, 𝑒𝑛−1}. Clearly, the vertices 𝑣1, 𝑣2, … , 𝑣𝑛−2, 𝑣𝑛in V−D forms a path and by the definition of semi total point graph the vertices 𝑒𝑛−2, 𝑒𝑛−1is adjacent to 𝑣𝑛−2, 𝑣𝑛in V−D. Thus <V−D> has no isolated vertices. Except 𝑣𝑛−1all other 

elements in D is exactly adjacent to two vertices in V−D, so the out degree of these elements in D is 

two. For 𝑣𝑛−1 in D, it is exactly adjacent with four vertices, hence the out degree is four. Thus, for any 

vertex 𝑢, 𝑣 ∈D, |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2.  So, D is the minimum co-total 2 - ODED set. Now,|𝐷| =𝑛 − 1.  Hence, 𝛾𝑐𝑡2𝑜𝑒(𝑇2(𝐶𝑛)) = 𝑛 − 1 for 𝑛 ≥ 3.  

Theorem 3.10 For any path  P𝑛, 𝛾𝑛𝑠2𝑜𝑒(T2(P𝑛)) = 𝑛 − 1 for  𝑛 ≥ 2 

Proof: Let P𝑛be a path for 𝑛 ≥ 2, Here V(P𝑛) = {𝑣1, 𝑣2, … , 𝑣𝑛} and E(P𝑛) = {𝑒1, 𝑒2, … , 𝑒𝑛−1}. Now, 

V(T2(P𝑛)) = {𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑒1, 𝑒2, … , 𝑒𝑛−1} be vertices of T2(P𝑛). Let D = {𝑒1, 𝑒2, … , 𝑒𝑛−1} be the 

minimal dominating set of T2(P𝑛). Then V−D = {𝑣1, 𝑣2, … , 𝑣𝑛}. The induced subgraph of <V−D> is 

the given P𝑛which has no isolated vertices .Hence two out degree of any vertex in D is 2. Clearly for 

any vertex 𝑢, 𝑣 ∈D, |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2. Therefore, D is the minimum co-total 2- ODED set. 

Hence,𝛾𝑐𝑡2𝑜𝑒(T2(P𝑛)) = 𝑛 − 1 for 𝑛 ≥ 2.\  

Theorem 3.11 For all Combo Graph  Pn+, γct2oe(T2(Pn+)) = 2n - 1 forn ≥ 2 

Proof: Let P𝑛+be a Combo Graph  for 𝑛 ≥ 2, Here V(P𝑛+) = {𝑣1, 𝑣2, … , 𝑣2𝑛} and E(P𝑛+) = {𝑒1, 𝑒2, … , 𝑒2𝑛−1}. Now, V(T2(P𝑛+)) = {𝑣1, 𝑣2, … , 𝑣2𝑛, 𝑒1, 𝑒2, … , 𝑒2𝑛−1} is the vertex set of T2(P𝑛+). Let 

D = {𝑒1, 𝑒2, … , 𝑒2𝑛−1} be the minimal dominating set of T2(P𝑛+). Then V−D = {𝑣1, 𝑣2, … , 𝑣2𝑛}. The 

subgraph induced by V−D is the given graph P𝑛+which has no isolated vertices and also each vertex in 

D is exactly adjacent to two vertices in V−D. Now, 𝑜𝑑𝐷(𝑢 )= 2 for any vertex𝑢, 𝑣 ∈D, |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2. Therefore, D is the minimum co-total 2 - ODED set. Hence 𝛾𝑐𝑡2𝑜𝑒(T2(P𝑛+)) 

= 2𝑛 − 1 for  𝑛 ≥ 2.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

899 https://internationalpubls.com 

Theorem 3.12 For all Fan graph  𝐹𝑛, γct2oe(T2(𝐹𝑛)) = 2n - 1 forn ≥ 2 

Proof: Let vertex set V(𝐹𝑛) = {𝑣, 𝑣1, 𝑣2, … , 𝑣𝑛} and edge set E(𝐹𝑛) = {𝑒1, 𝑒2, … , 𝑒2𝑛−1}. Now, V(T2(𝐹𝑛)) 

= {𝑣, 𝑣1, 𝑣2, … , 𝑣𝑛, 𝑒1, 𝑒2, … , 𝑒2𝑛−1} is the vertex set of T2(𝐹𝑛).  

Let D = {𝑒1, 𝑒2, … , 𝑒2𝑛−1} be the minimal dominating set of T2(𝐹𝑛). . Then V−D = {𝑣, 𝑣1, 𝑣2, … , 𝑣𝑛}. 

The induced subgraph of V−D is the given graph 𝐹𝑛 which has no isolated vertices and also each 

vertex in D is exactly adjacent to two vertices in V−D. Now, for any vertex𝑢, 𝑣 ∈D, |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2. Therefore, D is the minimum co-total 2 - ODED set. Hence 𝛾𝑐𝑡2𝑜𝑒(T2(𝐹𝑛)) = 2𝑛 − 1 for 𝑛 ≥ 2.  

Theorem 3.13 For any star graph  K1,𝑛, 𝛾𝑛𝑠2𝑜𝑒(T2(K1,𝑛)) = 𝑛 for 𝑛 ≥ 1 

Proof: Let K1,𝑛be a star graph for𝑛 ≥ 1with vertex set V(K1,𝑛) = {𝑣1, 𝑣2, … , 𝑣𝑛+1} and edge set E(K1,𝑛) 

= {𝑒1, 𝑒2, … , 𝑒𝑛}. Now, V(T2(K1,𝑛)) = {𝑣1, 𝑣2, … , 𝑣𝑛+1, 𝑒1, 𝑒2, … , 𝑒𝑛} be the vertices of T2(K1,𝑛). Let D 

= {𝑒1, 𝑒2, … , 𝑒𝑛} then V−D = {𝑣1, 𝑣2, … , 𝑣𝑛+1}. The induced subgraph of V−D is K1,𝑛, hence it has no  

vertices of degree zero. Also each vertices in D is exactly adjacent to two vertices in V−D. Hence for 

any vertex v ∈ 𝐷  then 𝑜𝑑𝐷(𝑣) = 2. Clearly, for any vertex 𝑢, 𝑣 ∈D, |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2. So, D 

is a minimum co-total 2 - ODED set. Hence 𝛾𝑐𝑡2𝑜𝑒(T2(K1,𝑛)) = 𝑛 for 𝑛 ≥ 1.   

Theorem 3.14 For any triangular snake graph nC3, γns2oe(T2(nC3)) = 2n for n ≥ 1 

Proof: Let nC3be a triangular snake graph for 𝑛 ≥ 1, with vertex set V(𝑛𝐶3) = {𝑣1, 𝑣2, … , 𝑣2𝑛+1} and 

edge set E(nC3) = {𝑒1, 𝑒2, … , 𝑒3𝑛}. Now, V(𝑇2(𝑛𝐶3)) = {𝑣1, 𝑣2, … , 𝑣2𝑛+1, 𝑒1, 𝑒2, … , 𝑒3𝑛} be the vertex 

set of T2(nC3). Let D = {𝑒2𝑛+1, … , 𝑒3𝑛, 𝑣𝑛+2, … , 𝑣2𝑛+1} be the minimal dominating set of T2(nC3) then 

V−D = {𝑣1, 𝑣2, … , 𝑣𝑛+1, 𝑒1, 𝑒2, … , 𝑒2𝑛}. Clearly, the vertices 𝑣1, 𝑣2, … , 𝑣𝑛+1in V−D forms a path and 

by the definition of semi total point graph the vertices 𝑒1, 𝑒2, … , 𝑒2𝑛 is adjacent to 𝑣1, 𝑣2, … , 𝑣𝑛+1in 

V−D. Thus <V−D> has no isolated vertices. The vertices 𝑒2𝑛+1, … , 𝑒3𝑛 in D is adjacent exactly two 

vertices in V−D, so the out degree is two. The remaining vertices 𝑣𝑛+2, … , 𝑣2𝑛+1of  D is  adjacent 

exactly four vertices in V−D, hence the out degree is four. Thus, for any vertex 𝑢, 𝑣 ∈D, |𝑜𝑑𝐷(𝑢) − 𝑜𝑑𝐷(𝑣)| ≤ 2. So, D is the minimum co-total  2 - ODED set. Now, |𝐷| = 2𝑛. 

Hence 𝛾𝑐𝑡2𝑜𝑒(T2(nC3)) = 2𝑛 for 𝑛 ≥1 

4. Conclusion 

In the next paper we study the bounds of co-total  2- ODED number and find the above  number for 

some new family of graphs. Also we like to extend the study to find the limitations and applications 

of the co-total  2- ODED  number. 

References 

1. Basavanagoud, B., & Tali, V. V. (2014). Equitable co-total domination number in graphs. 

International Journal of Scientific Research, 3(7), 301-305. 

2. Basavanagoud, B., & Hosamani, S. M. (2011). Connected semi-total point domination in graphs. 

International Journal of Science and Technology, 2(4), 116-125. 

3. Basavanagoud, B., & Malgham, S. H. (2010). Domination in semi-total point graph. Journal of 

Computer and Mathematical Sciences, 5, 598-605. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

900 https://internationalpubls.com 

4. Berge, C. (1962). Theory of graphs and its applications. Methuen, London. 

5. Harary, F. (1969). Graph theory. Addison-Wesley, Reading, MA. 

6. Kulli, V. R., Jankiram, B., & Radha R. Iyer. (1999). The co-total domination number of a graph. 

Journal of Discrete Mathematical Sciences & Cryptography, 2(2-3), 179-184. 

7. Mahesh, M. S., & Namasivayam, P. (2017). Connected two out-degree equitable domination of 

semi-total point graphs. Journal of Computer and Mathematical Sciences, 8(4), 133-138. 

8. Mathevan Pillai, K., Mahesh, M. S., Santiago Stephan, A., & Selvam, G. (2018). Some more results 

on non-split two out-degree equitable domination number. International Journal of Mechanical 

and Production Engineering Research and Development, 8(2), 361-369. 

9. Mathevan Pillai, K., Mahesh, M. S., & Selvam, G. (2016). Non-split two out-degree equitable 

domination number in graphs. Journal of Chemical and Pharmaceutical Sciences, 9(4), 2266-

2270. 

10. Ore, O. (1962). Theory of graphs. American Mathematical Society Colloquium Publications, 38. 

Providence, RI. 

11. Sahal, A., & Mathad, V. (2013). Two-out degree equitable domination in graphs. Transactions on 

Combinatorics, 2(3), 13-19.