The Upper -Edge Steiner Number of a Graph
A. Siva Jothi* and S. Robinson Chellathurai1 Department of Mathematics,
Scott Christian College, Nagercoil-629 003, Tamil Nadu, INDIA.
1Associate Professor, Department of Mathematics, Scott Christian College, Nagercoil-629 003, Tamil Nadu, INDIA.
Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, INDIA.
email: [email protected], robinchel@rediffmail.com
(Received on: August 16, 2018)
ABSTRACT
Let be a vertex of a connected graph and ⊆ ( ) such that ∉ . Then is called an -edge Steiner set of if every vertex of lies on some Steiner ∪ { } - tree of . The minimum cardinality of an -edge Steiner set of is defined as -edge Steiner number of and denoted by ( ). An -edge Steiner set is called a minimal -edge Steiner set of if no proper subset of is an -edge Steiner set of . The upper -edge Steiner number ( ) of is the maximum cardinality of a minimal -edge Steiner set of . The upper -edge Steiner number of certain classes of graph are determined. Some general properties satisfied by this concept are studied. For any positive integers and with 2 ≤ ≤ , there exists connected graph G such that ( ) = and ( ) = .
AMS Subject Classification: 05C12.
Keywords: Steiner number, edge Steiner number, x -edge Steiner number, upper – edge Steiner number.
1. INTRODUCTION
By a graph = ( , ), we mean a finite undirected connected graph without loops or multiple edges. The order and size of are denoted by and respectively. The distance ( , ) between two vertices and in a connected graph is the length of a shortest −
path in . An − path of length ( , ) is called an − geodesic. It is known that the distance is a metric on the vertex set of . For a vertex of , the eccentricity ( ) is the distance between and a vertex farthest from . The minimum eccentricity among the vertices of is the radius rad and the maximum eccentricity is its diameter, diam of . For basic graph theoretic terminology, we refer to Harary1,2. For a nonempty set of vertices in a connected graph , the Steiner distance ( ) of is the minimum size of a connected subgraph of containing . Necessarily, each such subgraph is a tree and is called a Steiner tree with respect to or a Steiner -tree. It is to be noted that ( ) = ( , ), when = { , }. If is an end vertex of a Steiner -tree, then ∈ . Also if < > is connected, then any Steiner -tree contains the elements of only. The Steiner distance was introduced in3. The set of all vertices of that lie on some Steiner -tree is denoted by ( ). If ( ) = , then is called a Steiner set for . A Steiner set of minimum cardinality is a minimum Steiner set or simply a -set of and this cardinality is the Steiner number ( ) of . If W is a Steiner set of and a cut vertex of , then lies in every Steiner -tree of and so ∪ { } is also a Steiner set of . The Steiner number of a graph was introduced in4 and further studied
in5,6,9. Let be a connected graph with at least 2 vertices. An edge Steiner set of is a set
⊆ ( ) such that every edge of is contained in a Steiner -tree. The edge Steiner number ( ) is the minimum cardinality of its edge Steiner sets and any edge Steiner set of cardinality ( ) is a minimum edge Steiner set of G. This concept was introduced in8. Any –Steiner set of cardinality ( ) is called a -set of . A vertex is an extreme vertex of a graph if the subgraph induced by its neighbors is complete. Let be a vertex of a connected graph and ⊂ V (G) such that ∉ . Then is called an -edge Steiner set of if every vertex of lies on some Steiner ∪ { } - tree of . The minimum cardinality of an -edge Steiner set of is defined as -edge Steiner number of and denoted by ( ). Any -edge Steiner set of cardinality ( ) is called an -set of G. This concept was studied in7. Throughout the following G denotes a connected graph with at least two vertices. The following theorem is used in sequel.
Theorem 1.1. [7] For the non-trivial tree with end vertices, ( ) =
− 1
2. THE UPPER -EDGE STEINER NUMBER OF A GRAPH
Definition 2.1. Let x be a vertex of a connected graph and W an x-edge Steiner set of .
⊆ ( ) such that ∉ . Then W is called a minimal x-edge Steiner set of if no proper subset of W is an x-edge Steiner set of . The upper x-edge Steiner number ( ) of is the maximum cardinality of a minimal x-edge Steiner set of .
Example 2.2. For the graph in Figure 2.1, the minimum x-edge Steiner sets and the upper x-edge Steiner numbers are given in the following Table 2.1.
G Figure 2.1
Table 2.1
Theorem 2.3. Each extreme vertex of a graph belongs to minimal x-edge Steiner set of . In particular, each end-vertex of belongs to every minimal x-edge Steiner set of .
Proof. Since every minimal x-edge Steiner set of is a x-edge Steiner set of . ∎
Theorem 2.4. Let x be a vertex of a connected graph with a cut-vertex of and W an minimal x-edge Steiner set of .
(i) If = , then every component of − contains an element of W.
(ii) If ≠ , then for each component C of − with
C, ∩ ≠ ∅.Proof. Let be a cut-vertex of , x be a vertex of and, W be an minimal x-edge Steiner set of .
(i) Let = . Suppose there exists a component say of − such that contains no vertex of W. By Theorem 2.3, W contains all the extreme vertices of and hence it follows that does not contain any extreme vertex of . Thus contains at least one edge say . Since every Steiner -tree T must have its end-vertex in W and is a cut-vertex of , it is clear that no Steiner -tree would contain the vertices and . This contradicts that W is an minimal x-edge Steiner set of .
(ii) Let ≠ . Suppose there exists a component say C of − with x
C such that W ∩ C = ∅. Then proceeding as in i), we get a contradiction. ∎Theorem 2.5. No cut-vertex of a connected graph belongs to any minimal x-edge Steiner set of .
Vertex x -sets Minimal x-edge Steiner sets (G)
{ , } { , , , } 4
{ , } { , , } 3
{ , } { , , , } 4
{ , , , } { , , , } 4
{ , } { , , , } 4
{ , , , } { , , , } 4
{ , } { , , } 3
Proof. Let be a cut-vertex of , be a vertex of and W be an x-edge Steiner set of . If = , then by the definition of the x-edge Steiner set,
. So let ≠ . Suppose that∈ . Let , , ..., (r ≥ 2) be the components of − . Let us assume that ∈ ( ).
Then by Theorem 2.4 (ii) each component Gi (2 ≤ ≤ ) contains an element of W. We claim that = − { } is also a x-edge Steiner set of . Since is a cut-vertex of , each Steiner - tree contains v. Now, since ∈ , it follows that each Steiner Wx - tree is also a Steiner - tree of . Thus is a Steiner set of such that ⊂ W which is a contradiction to W is a minimal x-edge Steiner set of . Hence the theorem. ∎
In the following we determine the minimal x -edge Steiner numbers of certain standard graphs.
Corollary 2.6. For the complete graph (p ≥ 2), ( ) = − 1 for every vertex x in . ∎ Corollary 2.7. For the non-trivial tree T with k end vertices,
(T) =
− 1 ∎ Theorem 2.8. For a complete bipartite graph = , (2 ≤ ≤ ) with bipartite set
= , , , , … , , , = { , , , , … , }
( ) = − 1 ∈
− 1 ∈
Proof. Let = { , , , , … , } and = { , , , , … , } be the bipartite sets of . Let ∈ and = − { }. Then it is clear that W is an -edge Steiner set of . We show that W is a minimal -edge Steiner set of . Let X be a subset of such that ⊊ . Then there exists at least one vertex, say such that ∉ . Now the vertex does not lie on a Steiner - tree of and so X is not an x-edge Steiner set of . If ⊊ , similarly we can prove that X is not a x-edge Steiner set of . If ⊊ ∪ , then <X> is connected. Hence X is not an x-edge Steiner set of . Therefore W is a minimal x-edge Steiner set of and so
(G) ≥ − 1. We show that (G) = − 1. Suppose that there is a minimal x-edge Steiner set Y of with | |≥ . Then ⊊ ∪ such that <Y> is connected. Hence Y is not an x-edge Steiner set of G. Thus (G) = m − 1. Similarly we can prove that (G) =
n − 1 if ∈ . ∎ Theorem 2.9. For positive integers , and > 2 with < ≤ 2 , there exists a
connected graph G with rad = r, diam = d and (G) = l for some vertex x in . Proof. When = 1, let = , then = 2 and then by Corollary 2.6, (G) = l for the internal vertex x in . Now, let r ≥ 2. Suppose that l ≥ 2. Construct a graph as follows. Let
: , , … , , be the cycle of order 2 and Let : , , , … , be a path of order − + 1. Let be the graph obtained from and by identifying in and in . Add l-2 new vertices , , … , to H and join each vertex (1 ≤ i ≤ l-2) to the vertex and join the vertices and and obtain the graph of Figure 2.2. Then rad = and diam = . Let = . Let W = { , , , … , , } be the set of all extreme vertices of . By Theorem 2.3, W belongs to every minimal x-edge
Steiner set of and so (G) ≥ l. It is clear that W is minimal x-edge Steiner set of so that ( ) = . ∎
Theorem 2.10. For every positive integers a and b with 2 ≤ ≤ b, there exists a connected graph such that ( ) = a and ( ) = b.
Proof. If a = b, take = , . Then by Corollary 2.6, ( ) = (G) = . If = 2, b > 3.
Let P: , , , , be the path on five vertices. Let be the graph by obtained from P by adding the new vertices , , … , and joining each (1 ≤ i ≤ b) with and . The graph is shown in Figure 2.3, Let x = . Let ≠ be a vertex of . Then { } is not an (G) ≥ 2. Let = { , }. Then W is an x-edge Steiner set of so that ( ) = 2. Next we prove that ( ) = b.
Let = { , , … , }. Then is an x -edge Steiner set of . We show that is a minimal x-edge Steiner set of . Let be any proper subset of . Then there exist at least one vertex say u ∈ T such that u ∉ . Now, assume that u = ℎ for some (1 ≤ ≤ ).
Then the vertex ℎ (1 ≤ ≤ b) does not lie on any Steiner -tree of . Now, assume that u
= . Then the vertex does not lie on any Steiner -tree of and so T is not a x-edge Steiner set of . Hence is a minimal x -edge Steiner set of so that ( ) ≥ b. We prove that ( ) = b. Let be a minimal x -edge Steiner set with | | = b. Then ⊈ and { , } ∉ . Suppose that ∉ for some , then does not lie on Steiner tree of . If ∉ for some i (1 ≤ ≤ 4) then does not lie on Steiner tree of . Therefore
is not a minimal x -edge Steiner set of so that ( ) = b.
G Figure 2.3
Let 2 < < . Let be the graph obtained in Figure 2.4 from the path on five vertices : , , , , by adding the new vertices ℎ , ℎ , … , ℎ and , , … , and joining each ℎ (1 ≤ ≤ − − 2) with and and also joining each (1 ≤ ≤ − 2) with . Let x = . Let = , , … , be the set of extreme vertices of . Let S be any x-edge Steiner set of . Then by Theorem 2.3, ⊆ . Then is not a x -edge Steiner set of and so (G) ≥ − 1. Also, it is easily verified that ∪ { }, where ∉ , is not a x -edge Steiner set of and so (G) ≥ a. Now, it is clear that W ∪ { , } is a x-edge Steiner set of so that (G) = a. Now, = ∪ {ℎ , ℎ , … , ℎ } is a x-edge Steiner set of .
We show that is a minimal x-edge Steiner set of . Let be any proper subset of . Then there exist at least one vertex say u ∈ such that ∉ . By Theorem 2.3, ≠ (1 ≤ ≤ − 2) Now, assume that u = ℎ for some (1 ≤ ≤ − − 2). Then the vertex ℎ (1 ≤ ≤ − − 2) does not lie on any Steiner -tree of . Now, assume that u = . Then the vertex does not lie on any Steiner -tree of and so W is not a -edge Steiner set of . Hence is a minimal x-edge Steiner set of so that ( ) ≥ b. Now, we show that there is no minimal x-edge Steiner set of with | | ≥ b + 1. Since | ( )| = b+2, there is no x-edge Steiner set with | | = b + 1. Thus ( ) = b. ∎
G Figure 2.4
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