Electronic Journal of Graph Theory and Applications 13 .
, 31Ae41 Steiner radial number resulting from various graph operations GurusamyOa .
Lakshmananb .
Ratha Jeyalakshmia .
Arockiarajc a Department of Mathematics.
Mepco Schlenk Engineering College.
Sivakasi 626005.
Tamil Nadu.
INDIA.
b PG and Research Department of Mathematics.
Thiagarajar College.
Madurai 625009.
Tamil Nadu.
INDIA.
c Departments of Mathematics,Government Arts & Science College.
Sivakasi 626124.
Tamil Nadu.
INDIA.
sahama2010@gmail.
com, lakshmsc2004@yahoo.
in, rratha@mepcoeng.
in, psarockiaraj@gmail.
O corresponding author Abstract The Steiner n-radial graph of a graph G on p vertices, denoted by SRn (G), has the vertex set as in G and any n.
O n O .
vertices are mutually adjacent in SRn (G) if and only if they are n-radial in G.
When G is disconnected, any n vertices are mutually adjacent in SRn (G) if not all of them are in the same component.
For the edge set of SRn (G), draw Kn corresponding to each set of n-radial vertices.
The Steiner radial number rS (G) of a graph G is the least positive integer n such that the Steiner n-radial graph of G is complete.
In this paper.
Steiner radial number has been determined for the line graph of any tree, total graph of any tree, complement of any tree, sum of two non-trivial trees and Mycielskians of some families.
For any pair of positive integers a, b Ou 3 with a O b, there exists a graph whose Steiner radial number is a and Steiner radial number of its line graph is b.
Keywords: n-radius.
Steiner n-radial graph.
Steiner radial number, line graph, total graph.
Mycielskian graph.
Mathematics Subject Classification : 05C12 DOI: 10.
5614/ejgta.
Introduction We consider finite undirected graphs without multiple edges and loops throughout this paper.
Let G be a graph on p vertices and S, a set of vertices of G.
The subgraph induced by S in G is denoted by S.
In .
, the Steiner distance (SD) of S in G denoted by dG (S), is defined as Received: 21 February 2020.
Revised: 19 December 2024.
Accepted: 27 January 2025.
Steiner radial number resulting from various graph operations Gurusamy et al.
the minimum number of edges in a connected subgraph of G that contains S.
Such a subgraph is necessarily a tree and is called a Steiner tree for S in G.
Steiner trees play a significant role in various aspects of image processing, such as segmentation, feature extraction, image registration, and compression.
They provide efficient methods for connecting and analyzing image features, thereby enhancing the capabilities of automated image analysis systems and improving the accuracy of image-based applications .
, .
The Steiner n-eccentricity en .
of a vertex v in a graph G is defined as en .
= max.
G (S) : S OI V (G) with v OO S and |S| = .
The n-radius radn (G) of G is defined as the smallest Stenier n-eccentricity among the vertices of G, and the n-diameter diamn (G) of G is the largest Steiner n-eccentricity.
The concept of SD was further developed in .
, .
In .
KM.
Kathiresan et al.
introduced the concept of Steiner radial number of a graph G.
Any n vertices of a graph G are said to be n-radial to each other if SD between them is equal to the n-radius of the graph G.
The Steiner n-radial graph of a graph G, denoted by SRn (G), has the vertex set as in G and n.
O n O .
vertices are mutually adjacent in SRn (G) if and only if they are n-radial in G.
If G is disconnected, any n vertices are mutually adjacent in SRn (G) if not all of them are in the same component.
For the edge set of SRn (G), draw Kn corresponding to each set of n-radial vertices.
When n = 2.
Steiner n-radial graph of G coincides with radial graph G.
For a pair of graphs G and H on p vertices, the least positive integer n such that SRn (G) O = H, is called the Steiner completion number of G over H.
When H = Kp , the Steiner completion number of G over His called as Steiner radial number of G.
The Steiner radial number rS (G) of a graph G is the least positive integer n such that the Steiner n-radial graph of G is complete.
The Steiner tree and Steiner radial number are valuable tools in network optimization across various domains, offering efficient solutions for connecting multiple points with minimal distance or cost.
Steiner radial graphs are valuable for visualizing complex networks such as social networks, communication networks, or biological networks .
, 23, .
The concept of graph operator has found various applications in chemical research .
, .
The line graph L(G) of a graph G has vertices corresponding to the edges of G and two vertices are adjacent in L(G) if their corresponding edges of G have a common end vertex .
Parameters of line graphs have been applied for the evaluation of structural complexity of molecular graphs and design of novel topological indices .
, .
The total graph T (G) of a graph G has vertex set V (G) O E(G) and two vertices of T (G) are adjacent whenever they are neighbours in G .
The vertices and edges of a graph are called Two elements of a graph are neighbours if they are either incident or adjacent.
Several properties of total graphs are investigated in .
, 2, 3, .
A tree is defined as a connected graph that contains no cycles, and various parameters associated with trees have been extensively studied .
, .
A vertex of degree zero in G is called an isolated vertex and a vertex of degree one is called a pendant vertex or a leaf.
An edge e in a graph G is called a pendant edge if it is incident with a pendant vertex.
A vertex of degree p Oe 1 is called an universal vertex or full degree The graph G obtained from K1,m and K1,n by joining their centers by an edge is called a bistar and is denoted by B.
, .
The join G = G1 G2 has V (G) = V (G1 ) O V (G2 ) and E(G) = E(G1 ) O E(G2 ) O .
v : u OO V (G1 ), v OO V (G2 )} if G1 and G2 are two graphs with disjoint vertex sets.
A vertex v in G is called peripheral if eccentricity of v is equal to diameter of Steiner radial number resulting from various graph operations Gurusamy et al.
A power graph Gk .
he k th power of a graph G) is the graph whose vertices are those of G and in which two distinct vertices are joined whenever the distance between them in G is at most k .
The complement G of a simple graph G is the simple graph with vertex set V (G) defined by uv OO E(G) if and only if uv is not in E(G) .
For a graph G = (V.
E), the Mycielskian of G is the graph AA(G) with vertex set V O V A O .
, where V A = .
iA : vi OO V } and edge set E O .
i vjA : vi vj OO E} O .
iA u : viA OO V A } .
The following notation can be found in .
The set of all connected graphs G for which r(G) = 1 and d(G) = 2 on p vertices denoted by F12 .
Now, we collect some useful results known for Steiner radial number of graphs from .
Theorem A .
For every tree T with m(= p Oe .
pendant vertices, rS (T ) = m 2.
Theorem B .
For any complete bipartite graph Kp1 ,p2 with p1 O p2 and p1 = 1, rS (Kp1 ,p2 ) = p1 1.
Theorem C .
If G is disconnected graph of order p Ou 3 but not totally disconnected, then SR3 (G) O = Kp .
Theorem D .
rS (G) = 2 if and only if G is either complete or totally disconnected.
For graph theoretic terminology, we follow .
Main Results In this section, we shall determine the Steiner radial number for the line graph of a graph, the total graph of a graph, the complement of a tree, and the Mycielskians of certain graphs.
In Observation 2.
1, we determine the Steiner radial number for the complement of a complete n-partite graph and illustrate the possible Steiner n-radial graphs in Example 2.
Observation 2.
For any complete n-partite graph other than complete graph, the Steiner radial number of its complement is 3.
Proof.
By Theorem C, the result follows.
Proposition 2.
If G OO F12 , then rS (G) = 3.
Proof.
Since G OO F12 .
R(G) O = G and hence SRn (G) O = G which is not complete.
Therefore, rS (G) Ou 3.
SD of any 3-element set having at least one full degree vertex is 2 and SD of any 3-element set in which none of them are adjacent or only 2 vertices are adjacent is 3.
Hence for any vertex v of degree p Oe 1 in G, e3 .
= 2 and for any vertex v of degree less than p Oe 1, e3 .
is either 2 or 3.
Hence 3-radius of G is 2.
Hence SR3 (G) O = Kp and hence rS (G) = 3.
Example 2.
All possible Steiner n-radial graphs of the line graph of K2,4 are given below:
Let G = K2,4 , and let its line graph, denoted by L(G), be shown in Figure 1.
If we let n = 2, rad2 (L(G)) = 2 and the sets S1 = .
11 , e22 }.
S2 = .
11 , e23 }.
S3 = .
11 , e24 }.
S4 = .
12 , e21 }.
S5 = .
12 , e23 }.
S6 = .
12 , e24 }.
S7 = .
13 , e21 }.
S8 = .
13 , e22 }.
S9 = .
13 , e24 }.
S10 = .
14 , e21 }.
S11 = .
14 , e22 } and S12 = .
14 , e23 } are the only sets of 2-radial vertices of L(G).
Hence the Steiner 2-radial graph of L(G) is obtained and shown in Figure 2.
If we take n = 3, rad3 (L(G)) = 3 and the sets S1 = .
11 , e12 , e23 }.
S2 = .
11 , e12 , e24 }.
S3 = .
11 , e13 , e22 }.
S4 = .
11 , e13 , e24 }.
S5 = .
11 , e14 , e22 }.
S6 = .
11 , e14 , e23 }.
S7 = .
11 , e22 , e23 }.
S8 = .
11 , e22 , e24 }.
S9 = .
11 , e23 , e24 }.
S10 = .
21 , e22 , e13 }.
S11 = .
21 , e22 , e14 }.
S12 = Steiner radial number resulting from various graph operations Gurusamy et al.
21 , e23 , e12 }.
S13 = .
21 , e23 , e14 }.
S14 = .
21 , e24 , e12 }.
S15 = .
21 , e24 , e13 }.
S16 = .
21 , e12 , e13 }.
S17 = .
21 , e12 , e14 } and S18 = .
21 , e13 , e14 } are the only sets of 3-radial vertices of L(G).
Hence the Steiner 3-radial graph of L(G) is obtained and shown in Figure 2.
If we let n = 4, rad4 (L(G)) = 4 and the sets S1 = .
11 , e22 , e23 , e24 }.
S2 = .
12 , e21 , e23 , e24 }.
S3 = .
13 , e21 , e22 , e24 }.
S4 = .
14 , e21 , e22 , e23 }.
S5 = .
11 , e12 , e13 , e24 }.
S6 = .
11 , e13 , e14 , e22 } and S7 = .
12 , e13 , e14 , e21 } are the only sets of 4-radial vertices of L(G).
Hence the Steiner 4radial graph of L(G) is obtained, which is isomorphic to SR3 (L(G)), as shown in Figure 3.
If we take n = 5, rad5 (L(G)) = 4 and the sets S1 = .
11 , e12 , e13 , e14 , e21 }.
S2 = .
11 , e12 , e22 , e23 , e24 }.
S3 = .
13 , e14 , e22 , e23 , e24 } and S4 = .
11 , e21 , e22 , e23 , e24 } are 5radial vertices of L(G).
Hence the Steiner 5-radial graph of L(G) is obtained and shown in Figure 3, which is isomorphic to K8 .
Figure 1.
G = K2,4 and its Line graph L(G) Figure 2.
Steiner 2-radial graph of L(G) Steiner radial number resulting from various graph operations Gurusamy et al.
Figure 3.
Steiner 3-radial graph of L(G) and Steiner 5-radial graph of L(G) Steiner radial number for Line graph of some graphs In the next theorem, we compute the Steiner radial number for tree.
Theorem 2.
For any tree T of order p other than star and bistar, rS (L(T )) = rS (T ).
When T is either a star or a bistar, rS (L(T )) is 2 and 3 respectively.
Proof.
Let T be any tree of order p other than star and bistar with m pendant vertices.
Let x1 , x2 , .
, xm be the m pendant edges of T .
Since L(T ) has p Oe 1 vertices, d(S) O p Oe 2 for any vertex subset S of L(T ).
For any vertex e in L(T ), en .
= p Oe 2 for n = m 1.
Let ei and ej be two non S pendant edges of T .
For any set X OI .
1 , x2 , .
, xm } with |X| = m Oe 1.
SD of the set .
i , ej } X is less than p Oe 2 and hence ei and ej are non-adjacent in Steiner .
-radial of L(T ).
Since .
-radius of L(T ) is p Oe 2 and .
i , ej , x1 , x2 , x3 , .
xm } has SD p Oe 2 in L(T ), the Steiner .
-radial of L(T ) is Kp .
Therefore, rS (L(T )) = m 2.
By Theorem A, the result follows.
When T is a star.
L(T ) is a complete graph on p Oe 1 vertices and hence rS (L(T )) = 2.
When T is a bistar B.
, .
on m n 2 vertices.
L(T ) is a graph obtained by identifying a vertex of Km 1 with a vertex of Kn 1 .
This shows that L(T ) OO F12 and by proposition 2.
1 , rS (L(T )) = 3.
In the next theorem, we compute the Steiner radial number for complete bipartite graph.
Proposition 2.
For any complete bipartite graph Kp1 ,p2 with p1 O p2 and p1 = 1, rS (L(Kp1 ,p2 )) = p2 1.
Proof.
Let .
1 , u2 , .
, up1 } and .
1 , v2 , .
, vp2 } be the two partitions of Kp1 ,p2 .
Let .
i,j : 1 O i O p2 , 0 O j O p1 Oe .
be the vertices of L(Kp1 ,p2 ).
For each vertex ei,j , n O p1 2n Oe 2 , if n p1 Oe 2, if p1 1 O n O p 2 en .
i,j ) = p1 p2 Oe 1, if p2 1 O n O p1 p2 Since any set of p2 vertices having the vertices e1,0 and e1,1 has SD less than n-radius, rS (L(Kp1 ,p2 )) > p2 .
By graph symmetry, if e1,0 is adjacent to all the remaining vertices in SRp2 1 (L(Kp1 ,p2 )), then the result follows.
By division algorithm, p2 = kp1 r.
The set S = .
i,j : 1 O i O kp1 r Steiner radial number resulting from various graph operations Gurusamy et al.
and j = .
Oe .
mod p1 } is a set of p2 vertices whose SD is p1 p2 Oe 2.
By adding any vertex ei,j OO V (L(Kp1 ,p2 )) Oe S.
SD of S O ei,j is p1 p2 Oe 1.
Hence e1,0 is adjacent all the vertices of SRp2 1 (L(Kp1 ,p2 )).
Steiner radial numbers highlight distinct variations between a graph and its line graph.
The following Theorem demonstrates the existence of graphs with specified Steiner radial numbers for both the original graph and its line graph Theorem 2.
For any pair of positive integers a, b Ou 3 with a O b there exists a graph whose Steiner radial number is a and Steiner radial number of its line graph is b.
Proof.
By taking p1 = a Oe 1 and p2 = b Oe 1 , using Theorem B and proposition 2.
2, the result Steiner radial number for Total graph of a tree In the upcoming theorem, we determine the Steiner radial number for the total graph of a tree.
Theorem 2.
For any tree T of order p with m(= p Oe .
pendant vertices, rS (T (T )) = 2m 2.
Proof.
Let T be a tree of order p with m number of pendant vertices and n number of pendent If n Ou 2m, then the n-eccentricities of all the vertices are equal.
Since a tree other than star has at least two non-pendant vertices.
SD of set of 2m or 2m 1 vertices with two non-pendant vertices is less than n-radius.
But the set of 2m 2 vertices having all pendant vertices and edges is equal to n-radius.
Hence the result follows.
In the next proposition, we compute the Steiner radial number for power graph.
Proposition 2.
Let G be a graph on p Ou 3 vertices with radius r.
Then 3, for r O k < d rS (G ) = 2, for r = d.
Proof.
When r O k < d.
Gk OO F12 and by proposition 2.
1, rS (Gk ) = 3.
When r = d.
Gk O = Kp and hence rS (G ) = 2.
In the next proposition, we compute the Steiner radial number for join of two graphs.
Proposition 2.
If G1 and G2 are non-trivial trees with p1 and p2 vertices respectively and none of them is a star graph, then rS (G1 G2 ) = min.
1 , p2 }.
Proof.
Assume p1 O p2 .
Let u1 , u2 , .
, up1 be the vertices of G1 and v1 , v2 , .
, vp2 be the vertices n, if n<p1 n, if n<p2 of G2 .
In G1 G2 , en .
i ) = nOe1, if nOup1 and en .
j ) = nOe1, if nOup2 for any i and j, 1 O i O p1 and 1 O j O p2 respectively.
If n < p1 , radn (G1 G2 ) = n and in this case SRn (G1 G2 ) is not complete on p1 p2 vertices.
No one of the vertex ui is adjacent to any one of vj , 1 O i O p1 and 1 O j O p2 .
Hence rS (G1 G2 ) Ou n 1 whenever n < p1 .
Suppose that n Ou p1 .
In this case radn (G1 G2 ) = n Oe 1.
Let X be a subset of V (G1 G2 ) with n-elements.
If X contains at least one element from each G1 and G2 , then SD of X is n Oe 1.
This implies that any two vertices in SRn (G1 G2 ) are adjacent.
Hence the result follows.
Steiner radial number resulting from various graph operations Gurusamy et al.
Steiner radial number for complement of a tree In the next result, we obtain a Steiner radial number for complement of a tree.
Theorem 2.
Let T be a tree other than star.
If T has either a pendant vertex which is adjacent to a vertex of degree 2 or a vertex and its neighbours all of degree 2, then rS (T ) = n.
Otherwise rS (T ) = d 2 where d is the minimum degree among the vertices of degree Ou 3.
Proof.
Let T be a tree and v be a vertex of T .
If v is a pendant vertex which is adjacent to a vertex of degree 2, then en .
T ) = n while n O 3 and n Oe 1 while n Ou 4.
If v is a vertex of degree 2 whose neighbours are also of degree 2, then en .
T ) = n while n O 3 and n Oe 1 while n Ou 4.
T has a vertex and its neighbours all of degree 2, then radn (T ) = n while n O 3 and n Oe 1 while n Ou 4.
In case of n O 3.
SRn (T ) is not complete.
Since there is no set of n-elements containing at least two peripheral vertices of T with SD n.
When n = 4, radn (T ) = 3 and SRn (T ) is complete.
Assume that T has neither a pendant vertex adjacent to a vertex of degree 2 nor a vertex and its neighbours all of degree 2.
In this case every vertex v of T is adjacent to at least one vertex say u of degree Ou 3.
Among all the vertices of T with degree at least 3, let v has the smallest degree d.
Any set S having the vertex v and a non-neighbour of v is of SD |S| Oe 1.
Any set S having n,theif vertices nO di 1 from N .
is of SD |S|.
Hence for each vertex ui of T with degree di , en .
T ) = nOe1, if nOu n, if nO di 1 and for each vertex w OO N .
i ), en .
T ) = nOe1, if nOu di 2 .
This implies that radn (T ) = n if n O d 1 and n Oe 1 if n Ou d 2.
In T , any set S on .
-elements having at least two peripheral vertices is of SD d.
Hence SRd 1 (T ) is not complete.
Take n = d 2.
Since any set S of size d 2 having any two pendant vertices of T and at least two peripheral vertices of T is of SD d 1 in T .
SRn (T ) is complete.
Hence the result follows.
If T is a star on p vertices, then T is KpOe1 O K1 and by Theorem C, rS (T ) = 3.
Corollary 2.
For any positive integer n, rS (Pn ) = 4, if n Ou 4, n, if n = 1, 2, 3.
Proof.
By 2.
4, the result is true for n Ou 5.
When n = 4,Pn O = P4 and hence by Theorem A.
O rS (P4 ) = 4.
When n = 3.
Pn = P2 O K1 and hence by Theorem C, rS (P3 ) = 3.
When n = 2.
Pn is totally disconnected and hence by Theorem D, rS (P2 ) = 2.
When n = 1, the result is trivial.
Corollary 2.
For any positive integers m1 and m2 with m1 O m2 , rS (Bm1 ,m2 ) = m1 3 Steiner radial number for Mycielskians of some standard graphs In the following results, we shall explore the computation of the Steiner radial number for the Mycielskians of a certain family of graphs.
Theorem 2.
For any complete bipartite graph Km,n , rS (AA(Km,n )) = 3.
Proof.
Let A=.
1 , a2 , .
, am } and B=.
1 , b2 , .
, bn } be the two partitions of vertex set V and let E be edge set of Km,n .
Then the Mycielskian of Km,n is AA(Km,n ) with vertex set V A =A O B O AA O B A O .
, where AA =.
Ai : ai OO A} and B A =.
Ai : bi OO B} and edge set E A =E O .
i bAj : ai bj OO E} O .
i aAj : bi aj OO E} O .
Ai u : aAi OO A} O .
Ai u : bAi OO B}.
Since AA(Km,n ) is not isomorphic to Steiner radial number resulting from various graph operations Gurusamy et al.
K2.
1 , by theorem D, rS (AA(Km,n )) Ou 3.
For any vertex v in AA(Km,n ), e3 .
= 3 and hence 3Oeradius is 3.
The Steiner 3-radial graph of AA(Km,n ) is SR3 (AA(Km,n )) has the vertex set V A and edge set can be obtained from the following cases.
Case 1.
Each of the sets .
i , aj , .
, .
i , bj , .
, .
Ai , aAj , ai } and .
Ai , bAj , bi } for all 1 O i O m, 1 O j O n with i = j have SD 3 respectively.
Hence the subgraphs [A], [B], [AA ] and [B A ] are complete in SR3 (AA(Km,n )).
Case 2.
Each of the sets .
i , bj , .
, .
i , aAi , .
, .
i , bAj , aAi }, .
j , bAj , .
, .
j , bAj , aAi } and .
Aj , bAk , aAi } for all 1 O i O m, 1 O j, k O n with j = k have SD 3 respectively.
Therefore, all the elements in A.
AA .
B A and .
are mutually adjacent in SR3 (AA(Km,n )).
Thus Steiner 3-radial of AA(Km,n ) is K2.
Theorem 2.
AoLet G be a graph of order n with diameter at most 2 and OI(G) = n Oe 1.
Then rS (AA(G)) = 3.
Proof.
Let V =.
1 , v2 , .
, vn } be the vertex set and E be the edge set of G.
Then the Mycielskian of G is AA(G), has the vertex set V O V A O .
, where V A = .
iA : vi OO V } and edge set E O .
i vjA : vi vj OO E} O .
iA u : viA OO V A }.
Since OI(G) = n Oe 1, there exists a vertex v in G such that deg.
= n Oe 1, for that v, e3 .
= 3 in AA(G).
Hence 3Oeradius of AA(G) is 3.
Case 1.
Assume that diameter of G is 1.
Then each of the sets .
i , vj , .
, .
i , viA , .
i , viA , vjA } in AA(G) have SD 3 and forms a K3 in the corresponding Steiner 3Oeradial graph.
Thus SR3 (AA(G)) = K2n 1 .
Case 2.
Assume that diameter of G is 2.
Then each of the sets .
i , vj , .
, .
iA , vjA , vkA } and .
i , viA , .
have SD 3.
Hence in SR3 (AA(G)) the subgraph [V ] O .
and [V A ] O .
are complete.
Hence to prove SR3 (AA(G)) is complete, it is enough to show that every vertex in V is adjacent to all the vertices in V A .
Subcase a.
If vi is a full degree vertex in G, the sets .
i , viA , vjA } having SD 3.
Thus the subgraph [V O V A ] is complete in SR3 (AA(G)).
Hence the result follows.
Subcase b.
If vi is not a full degree vertex, then there exists a vertex vj not adjacent with vi .
Since diameter of G is 2, there exists a vertex vk common to vi and vj .
Then the sets .
i , vkA , vjA } having SD 3 and hence the subgraph [V O V A ] is complete in SR3 (AA(G)).
Hence the result follows.
Theorem 2.
For any path Pn with n Ou 7 vertices.
SR3 (AA(Pn )) = K2n 1 Oe K1,n Proof.
Let V =.
0 , v1 , v2 , .
, vnOe1 } be the vertex set and E be the edge set of Pn .
Then the Mycielskian of Pn is AA(Pn ), has the vertex set V O V A O .
, where V A = .
iA : vi OO V } and edge set E O .
i vjA : vi vj OO E} O .
iA u : viA OO V A }.
Since AA(Pn ) is not isomorphic to K2n 1 , by theorem D, rS (AA(Pn )) Ou 3.
For any vertex in AA(Pn ),e3 .
i ) = 6, e3 .
iA ) = 5 and e3 .
= 4 for all 0 O i O nOe1 and hence 3Oeradius is 4.
SR3 (AA(Pn )) has the vertex set as in AA(Pn ) and edge set can be obtained from the following cases.
Case 1.
The sets .
i , vj , u : d.
i , vj ) Ou 3 in Pn } and .
i , vj , vk : d.
i , vj ) = 2 and d.
i , vk ) = 2 or d.
j , vk ) = 2 in Pn } have SD 4 and forms K3 in SR3 (AA(Pn )).
Let vk be the eccentric vertex of vi in Pn .
Then .
i , vj , vkA : d.
i , vj ) = 1 in Pn } has SD 4.
Hence in SR3 (AA(Pn )), the subgraph [V ] O .
is complete.
Case 2.
For any two vertices viA and vjA in V A , there exists a vertex vk OO V such that d.
jA , vk ) = 2 Steiner radial number resulting from various graph operations Gurusamy et al.
or d.
iA , vk ) = 2.
Hence the sets .
iA , vjA , vk } forms K3 in SR3 (AA(Pn )).
Thus the subgraph [V A ] is complete in SR3 (AA(Pn )).
Case 3.
Let vk be the eccentric vertex of vi in Pn .
Then the sets .
i , vjA , vk } and .
i , vjA , vkA } have SD 4 when vi and vj are adjacent and non-adjacent respectively.
Hence the subgraphs [V ] and [V A ] are mutually adjacent in SR3 (AA(Pn )).
Case 4.
For any two vertices .
iA , .
, there is no vertex w in AA(Pn ) such that d(.
iA , u, .
) = 4.
Hence the vertices viA and u are not adjacent in SR3 (AA(Pn )).
Hence the result follows.
Observation 2.
For any path Pn , rS (AA(Pn )) = 3, for 1 O n O 4.
Theorem 2.
For any path Pn with n Ou 5 vertices, rs (AA(Pn )) = UO n3 UO 2.
Proof.
Let V = .
1 , v2 , v3 , .
, vn } be the vertex set and E be the edge set of Pn .
Then the Mycielskian of Pn is AA(Pn ), which has the vertex set V O V A O .
where V A = .
iA : vi OO V and edge set EO.
i vjA : vi vj OO E}O.
iA u : viA OO V A }.
For each vertex w OO V OV A , eUO n3 UO 1 .
= 2UO n3 UO 1 and eUO n3 UO 1 .
= 2UO n3 UO.
Hence (UO n3 UO .
Oe radius is 2UO n3 UO.
Consider the set S with UO n3 UO vertices from V such that d.
i , vj ) Ou 3 if vi , vj OO S.
Then the set S Oe .
i } O .
, viA } with (UO n3 UO .
vertices has SD 2(UO n3 UO Oe .
1 < 2UO n3 UO.
Therefore, u is not adjacent with viA in Steiner (UO n3 UO .
- radial graph of AA(Pn ).
Hence, rs (AA(Pn )) = UO n3 UO 2.
Now eUO n3 UO 2 .
= 2UO n3 UO 1 and eUO n3 UO 2 .
= 2UO n3 UO 2 where w OO V O V A and hence (UO n3 UO .
- radius is 2UO n3 UO 1.
Consider the set S A = .
O .
jA } O Si where Si = .
i , vi 3 , vi 6 , .
} with |Si | = UO n3 UO for i = 1, 2, 3 and vj OO Si .
Clearly.
SD of the set S A is 2UO n3 UO 1.
Hence it gives u is adjacent to all vertices of V and V A and all vertices of V is adjacent to V A .
Hence, to prove SRUO n3 UO 2 (AA(Pn )) = K2n 1 , it is enough to prove the subgraphs [V ] and [V A ] are complete in SRUO n3 UO 2 (AA(Pn )).
Case 1.
For every vertices vj OO Si and vj OO V, the set Si O .
j , vjA } has SD 2UO n3 UO 1.
Hence subgraph [V ] is complete in SRUO n3 UO 2 (AA(Pn )).
Case 2.
Let SiA = .
A OO V A : v OO Si OI V }.
Subcase a.
For any vjA , vkA OO SiA , the sets Si Oe .
j } O .
jOe1 , vjA , vk } or Si Oe .
j } O .
j 1 , vjA , vkA } have SD 2UO 3 UO 1.
Subcase b.
For any vjA and vkA in different SiA , then clearly either vk 1 or vkOe1 in SiA .
Hence, the sets A A } have SD 2UO n3 UO 1.
Therefore, the Si Oe .
k 1 } O .
jA , vkA , vk 1 } or Si Oe .
kOe1 } O .
jA , vkA , vkOe1 above two subcases make [V ] is complete in SRUO n3 UO 2 (AA(Pn )).
Hence the result follows.
Acknowledgement The authors are grateful to the Editor of the journal and the anonymous reviewers for their valuable comments and suggestions that led to a considerable improvement of the paper over its original version.
References