ISSN 2319-8133 (Online)
(An International Research Journal), www.compmath-journal.org
The Upper total Edge-to-Vertex Steiner Number of a Graph
S. Ancy Mary
1and S. Joseph Robin
2Register Number-12416,
Scott Christian College, Nagercoil-629 003, INDIA.
Associate Professor, Department of Mathematics,
Scott Christian College, Nagercoil-629 003, INDIA.
Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, INDIA.
(Received on: November 23, 2018)
ABSTRACT
Let 𝐺 = (𝑉, 𝐸) be a connected graph with at least 3 vertices. A set 𝑊 ⊆ 𝐸 is called an edge-to-vertex Steiner set if every vertex of 𝐺 lies on a Steiner 𝑊𝑒𝑣-tree of 𝐺. The edge-to-vertex Steiner number 𝑠𝑒𝑣(𝐺) of 𝐺 is the minimum cardinality of its edge-to-vertex Steiner sets and any edge-to-vertex Steiner sets of cardinality 𝑠𝑒𝑣(𝐺) is a minimum edge-to-vertex Steiner set of 𝐺. An upper edge-to-vertex Steiner set 𝑊 in a connected graph 𝐺 is called a minimal edge-to-vertex Steiner set of 𝐺 if no proper subset of 𝑆 is an edge-to-vertex Steiner set of 𝐺. The upper edge-to-vertex Steiner number of 𝐺 is the maximum cardinality of a minimal edge-to-vertex Steiner set of 𝐺. A total edge-to-vertex Steiner set 𝑊 in a connected graph 𝐺 is called a minimal total edge-to-vertex Steiner set of 𝐺 if no proper subset of 𝑊 is a total edge- to-vertex Steiner set of 𝐺. The upper total edge-to-vertex Steiner number of 𝐺 is the maximum cardinality of a minimal total edge-to-vertex Steiner set of 𝐺. It is denoted by 𝑠𝑡𝑒𝑣+ (𝐺). The upper total edge-to-vertex Steiner number of certain classes of graph are determined. Some general properties satisfied by this concept are studied. For any positive integers a and b with 2 ≤ 𝑎 ≤ 𝑏, there exists a connected graph 𝐺 such that 𝑠𝑡𝑒𝑣(𝐺) = a and 𝑠𝑡𝑒𝑣+ (G) = b.
AMS Subject Classification: 05C12.
Keywords: Edge-to-vertex Steiner number, upper edge-to-vertex Steiner number, upper total edge-to-vertex Steiner number.
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 𝑝 and 𝑞 respectively. The distance 𝑑(𝑢, 𝑣) between two vertices 𝑢 and 𝑣 in a connected graph 𝐺 is the length of a shortest 𝑢 − 𝑣 path in 𝐺. An 𝑢 − 𝑣 path of length 𝑑(𝑢, 𝑣) is called an 𝑢 − 𝑣 geodesic. It is known that the distance is a metric on the vertex set of 𝐺. For a vertex 𝑣 of 𝐺, the eccentricity 𝑒(𝑣) is the distance between 𝑣 and a vertex farthest from 𝑣. The minimum eccentricity among the vertices of 𝐺 is the radius, 𝑟𝑎𝑑𝐺 and the maximum eccentricity is its diameter, 𝑑𝑖𝑎𝑚𝐺 of 𝐺. For basic graph theoretic terminology, we refer to Harary
1,2. For a nonempty set 𝑊 of vertices in a connected graph 𝐺, the Steiner distance 𝑑(𝑊) of 𝑊 is the minimum size of a connected subgraph of 𝐺 containing 𝑊. Necessarily, each such subgraph is a tree and is called a Steiner tree with respect to 𝑊 or a Steiner 𝑊 - tree. It is to be noted that 𝑑(𝑊) = 𝑑(𝑢, 𝑣), when 𝑊 = {𝑢, 𝑣}. If 𝑣 is an end vertex of a Steiner 𝑊-tree, then 𝑣 ∈ 𝑊. Also if < 𝑊 > is connected, then any Steiner 𝑊-tree contains the elements of 𝑊 only. The Steiner distance was introduced in
3. The set of all vertices of 𝐺 that lie on some Steiner W-tree is denoted by 𝑆(𝑊).
If 𝑆(𝑊) = 𝑉, then W is called a Steiner set for 𝐺. A Steiner set of minimum cardinality is a minimum Steiner set or simply a s-set of 𝐺 and this cardinality is the Steiner number 𝑠(𝐺) of 𝐺. If 𝑊 is a Steiner set of 𝐺 and 𝑣 a cut vertex of 𝐺, then v lies in every Steiner 𝑊-tree of 𝐺 and so 𝑊 ∪ {𝑣} is also a Steiner set of 𝐺. The Steiner number of a graph was introduced in
4and further studied in
5,6,9. Let 𝐺 be a connected graph with at least 2 vertices. An edge Steiner set of 𝐺 is a set W ⊆ V(G) such that every edge of 𝐺 is contained in a Steiner 𝑊-tree. The edge Steiner number 𝑠
1(𝐺) is the minimum cardinality of its edge Steiner sets and any edge Steiner set of cardinality 𝑠
1(𝐺) is a minimum edge Steiner set of 𝐺. This concept was introduced in
8. Let 𝐺 = (𝑉, 𝐸) be a connected graph with at least 3 vertices. A set 𝑊 ⊆ 𝐸 is called an edge-to-vertex Steiner set if every vertex of 𝐺 lies on a Steiner 𝑊
𝑒𝑣-tree of 𝐺. The edge-to-vertex Steiner number 𝑠
𝑒𝑣(𝐺) of 𝐺 is the minimum cardinality of its edge-to-vertex Steiner sets and any edge-to-vertex Steiner sets of cardinality 𝑠
𝑒𝑣(𝐺) is a minimum edge-to- vertex Steiner set of 𝐺. Throughout the following 𝐺 denotes a connected graph with at least two vertices. The following theorem is used in sequel.
Theorem 1.1.
7For any connected graph 𝐺 of size 𝑞 ≥ 2, 𝑠
𝑡𝑒𝑣(𝐺) = q if and only if 𝐺 is a star.
Theorem 1.2.
7Let 𝐺 be a connected graph and 𝑊 be a 𝑠
𝑡𝑒𝑣-set of 𝐺. Then no cut edge of G which is not an end-edge of 𝐺 belongs to W.
Theorem 1.3.
7Every end-edge of a connected graph 𝐺 belongs to every edge-to-vertex Steiner set of 𝐺.
Theorem 1.4.
7Let 𝐺 be a connected graph and 𝑊 be a minimal edge-to-vertex Steiner set of
𝐺. Then no cut- edge of 𝐺 which is not an end-edge of 𝐺 belongs to 𝑊.
2. THE UPPER TOTAL EDGE-TO-VERTEX STEINER NUMBER OF GRAPH Definition 2.1. A total edge-to-vertex Steiner set 𝑊 in a connected graph 𝐺 is called a minimal total edge-to-vertex Steiner set of 𝐺 if no proper subset of 𝑊 is a total edge-to-vertex Steiner set of 𝐺. The upper total edge-to-vertex Steiner number of 𝐺 is the maximum cardinality of a minimal total edge-to-vertex Steiner set of 𝐺. It is denoted by 𝑠
𝑡𝑒𝑣+(𝐺).
Example 2.2. For the graph 𝐺 given in Figure 2.1, 𝑊
1= {𝑣
1𝑣
2, 𝑣
2𝑣
3, 𝑣
5𝑣
6, 𝑣
4𝑣
5} is a total edge-to-vertex Steiner set of 𝐺 so that 𝑠
𝑡𝑒𝑣(𝐺) = 4.
The set 𝑊
′= {𝑣
1𝑣
2,𝑣
6𝑣
3, 𝑣
5𝑣
4, 𝑣
2𝑣
6, 𝑣
6𝑣
4} is an upper edge-to-vertex Steiner set of 𝐺 and it is clear that no proper subset of 𝑊
′is a minimal edge-to-vertex Steiner set of 𝐺 and so 𝑠
𝑡𝑒𝑣+(𝐺)
≥ 5. It is easily verified that there is no minimal total edge-to-vertex Steiner set of cardinality greater than 5. Hence 𝑠
𝑡𝑒𝑣+(𝐺) = 5. ∎
Figure 2.1
Theorem 2.3. For a connected graph 𝐺, 2 ≤ 𝑠
𝑡𝑒𝑣(𝐺) ≤ 𝑠
𝑡𝑒𝑣+(𝐺) ≤ 𝑞.
Proof. Any total edge-to-vertex Steiner set needs at least two edges, and so 𝑠
𝑡𝑒𝑣(𝐺) ≥ 2. Since every minimal total edge-to-vertex Steiner set is a total edge-to-vertex Steiner set, 𝑠
𝑡𝑒𝑣(𝐺) ≤ 𝑠
𝑡𝑒𝑣+(𝐺). Also, Since 𝐸(𝐺) is a total edge-to-vertex Steiner set of 𝐺, it is clear that 𝑠
𝑡𝑒𝑣+(G) ≤ 𝑞. Thus 2 ≤ 𝑠
𝑒𝑣(G) ≤ 𝑠
𝑒𝑣+(G) ≤ 𝑞. ∎
Remark 2.4. The bound in Theorem 2.3 are sharp. For the path 𝐺 = 𝑃
3, 𝑠
𝑡𝑒𝑣(𝐺) = 2. For the double star 𝐺, 𝑠
𝑡𝑒𝑣(G) = 𝑠
𝑡𝑒𝑣+(𝐺) and for the star 𝐺 = 𝐾
1,𝑞, 𝑠
𝑡𝑒𝑣+(G) = 𝑞. Also, all the inequalities in the Theorem 2.3 are strict. For the graph 𝐺 given in Figure 2.1, 𝑠
𝑡𝑒𝑣(G) = 4, 𝑠
𝑡𝑒𝑣+(G) = 5 and 𝑞 = 8. ∎
Theorem 2.5. For a connected graph 𝐺, 𝑠
𝑒𝑣(G) = 𝑞 if and only if 𝑠
𝑒𝑣+(G) = 𝑞.
Proof. Let 𝑠
𝑡𝑒𝑣(G) = 𝑞. Then 𝑆 = 𝐸(𝐺) is the unique minimal total edge-to-vertex Steiner set of 𝐺. Since no proper subset of 𝑆 is a total edge-to-vertex Steiner set of 𝐺, it is clear that 𝑆 is the unique minimum total edge-to-vertex Steiner set of 𝐺 and so 𝑠
𝑡𝑒𝑣+(𝐺) = q. The converse follows from Theorem 2.3. ∎
Corollary 2.6. For the connected graph 𝐺, the following are equivalent.
a) 𝑠
𝑡𝑒𝑣(G) = 𝑞, b) 𝑠
𝑡𝑒𝑣+(G) = 𝑞, c) 𝐺 = 𝐾
1,𝑞. ∎
Theorem 2.3. For a conne
Theorem 2.7. For the complete graph 𝐺 = 𝐾
𝑝(p ≥ 4), 𝑠
𝑡𝑒𝑣+(𝐾
𝑝) = 𝑝 − 1.
Proof. Let 𝑊 be any set of 𝑝 − 1 adjacent edges of 𝐾
𝑝incident at a vertex, say x. Since every vertex of 𝐾
𝑝lies on Steiner 𝑊
𝑒𝑣-tree of 𝐺, it is clear that 𝑊 is a total edge-to-vertex Steiner set of 𝐺. If 𝑊 is not a minimal total edge-to-vertex Steiner set of 𝐺, then there exist a proper subset of 𝑊
′of 𝑊 such that 𝑊
′is an upper total edge-to-vertex Steiner set of 𝐺. There exists at least one vertex, say 𝑦 of 𝐾
𝑝such that 𝑦 is not incident with any edges of 𝑊
′. Hence 𝑦 does not belongs to any Steiner 𝑊
′𝑒𝑣-tree of 𝐺, Which is a contradiction. Hence 𝑊 is a minimal total edge-to-vertex Steiner set of 𝐺. Therefore 𝑠
𝑡𝑒𝑣+(𝐾
𝑝) ≥ 𝑝 − 1. Suppose that there exists a minimal total edge-to-vertex Steiner set of S such that |𝑆| ≥ 𝑝. Since S contains atleast 𝑝 edges, < 𝑆 > contains atleast one cycle. Let 𝑆
′= 𝑆 – {e}, where 𝑒 is an edge of a cycle which lies < 𝑆 >. It is clear that 𝑆
′is a total edge-to-vertex Steiner set with 𝑆
′⊂ S, Which is a contradiction. Therefore 𝑠
𝑡𝑒𝑣+(𝐾
𝑝) = 𝑝 − 1. ∎
Theorem 2.8. For the complete bipartite graph G = 𝐾
𝑚,𝑛(2 ≤m ≤ n), 𝑠
𝑡𝑒𝑣+(G) = m + n - 2.
Proof. Let 𝑈 = {𝑢
1,𝑢
2,𝑢
3,,𝑢
4,… , 𝑢
𝑚}, and 𝑉 = {𝑣
1,𝑣
2,𝑣
3,,𝑢
4,… , 𝑣
𝑛} be the bipartite
sets of 𝐺. Let 𝑊
𝑖= {𝑢
𝑖𝑣
1, 𝑢
𝑖𝑣
2, … , 𝑢
𝑖𝑣
𝑛−1, 𝑢
2𝑣
𝑛,,… , 𝑢
𝑖−1𝑣
𝑛, 𝑢
𝑖+1𝑣
𝑛, … , 𝑢
𝑚𝑣
𝑛}, (1 ≤ i ≤ m), 𝑆
𝑗= {𝑢
1𝑣
𝑗, 𝑢
2𝑣
𝑗, … , 𝑢
𝑚−1𝑣
𝑗,𝑢
𝑚𝑣
1,𝑢
𝑚𝑣
2,,… , 𝑢
𝑚𝑣
𝑖−1, 𝑢
𝑚𝑣
𝑖+1, … , 𝑢
𝑚𝑣
𝑛}, (1 ≤ j ≤ n) and 𝑊
𝑘= {𝑢
1𝑣
1, 𝑢
2𝑣
2, … , 𝑢
𝑚−1𝑣
𝑚−1,𝑢
𝑚𝑣
𝑚, 𝑢
𝑚𝑣
𝑚+1,,… , 𝑢
𝑚𝑣
𝑛} with |𝑊
𝑖| = |𝑆
𝑗| = m+ n - 2 and
|𝑊
𝑘| = n. It is easily verified that any minimal total edge-to-vertex Steiner set of 𝐺 is of the form either 𝑊
𝑗or 𝑆
𝑗or 𝑊
𝑘. Since no proper subset of 𝑊
𝑖(1 ≤ 𝑖 ≤ 𝑚), 𝑆
𝑗(1 ≤ 𝑗 ≤ 𝑛) and 𝑊
𝑘is an edge-to-vertex Steiner set of 𝐺, it is follows that 𝑠
𝑡𝑒𝑣+(G) = m + n - 2. ∎ Theorem 2.9. For any non-trivial tree 𝑇 with 𝑘 end vertices, 𝑠
𝑡𝑒𝑣+(𝑇) ≤ 2𝑘.
Proof. Let T be a tree with 𝑘 end edges. Let {𝑒
1, 𝑒
2, … , 𝑒
𝑘} be the set of end edges of T and 𝑓
𝑖is the adjacent edge of 𝑒
𝑖(1 ≤ 𝑖 ≤ 𝑘).
Then 𝑊 = {𝑒
1, 𝑒
2, … , 𝑒
𝑘, 𝑓
1, 𝑓
2, … , 𝑓
𝑘} is a total edge-to-vertex Steiner set of 𝐺 so that 𝑠
𝑡𝑒𝑣+(𝑇)
≤ 2𝑘. ∎
Corollary 2.10. For a path 𝑠
𝑡𝑒𝑣+(𝐺) = 4. ∎
Theorem 2.11. For every positive integers a and b with 2 ≤ 𝑎 ≤ 𝑏, there exists a connected graph 𝐺 such that 𝑠
𝑡𝑒𝑣(G) = 𝑎 and 𝑠𝑡𝑒𝑣+ (G) = 𝑏.Proof. Let 𝐾 ̅̅̅ = {𝑥, 𝑦}. Let 𝐺 be the graph obtained from 𝐾
2̅̅̅ by adding new vertices
2{𝑧, 𝑤, 𝑧
1, 𝑧
2, . . . , 𝑧
𝑎−2, 𝑥
1, 𝑥
2, 𝑥
3, 𝑥
4, … , 𝑥
𝑏−𝑎−4} by join each 𝑧
𝑖(1 ≤ 𝑖 ≤ 𝑎 −1) with 𝑦 and join each 𝑥
𝑖(1 ≤ 𝑖 ≤ 𝑏 − 𝑎 −4) with 𝑥 and 𝑦. The graph 𝐺 is shown in Figure 2.2.
First we show that 𝑠
𝑡𝑒𝑣(𝐺) = 𝑎. Let 𝑍 = {𝑦𝑧
1, 𝑦𝑧
2, . . . , 𝑦𝑧
𝑎−2} be the set of end edges in 𝐺.
By Theorem 1.3, Z is a subset of every total edge-to-vertex Steiner set of 𝐺. It is clear that 𝑍 is not a total edge-to-vertex Steiner set of 𝐺. Let 𝑇 be the total edge-to-vertex Steiner set of 𝐺.
It is easily and shown that 𝑍 ∪ {𝑒}, 𝑒 ∉ 𝑍 is not a total edge-to-vertex Steiner set of 𝐺 and so that 𝑠
𝑡𝑒𝑣(𝐺) = 𝑎. Then 𝑍
1is a total edge-to-vertex Steiner set of 𝐺, so that 𝑠
𝑡𝑒𝑣(𝐺) = 𝑎.
Next we show that 𝑠
𝑡𝑒𝑣+(𝐺) = 𝑏.
Let 𝑇 = 𝑍 ∪ {𝑥𝑥
2, 𝑥𝑥
3, … 𝑥
𝑏−𝑎−4,… , 𝑥
1𝑧, 𝑦𝑧, … , 𝑦𝑥
𝑏−𝑎−4, 𝑤𝑥
𝑏−𝑎−4}, is a total edge-to-vertex Steiner set of 𝐺 and so 𝑠
𝑡𝑒𝑣+(𝐺) = b. We show that 𝑇 is minimal total edge-to-vertex Steiner set of 𝐺. Let 𝑊 be any proper subset of 𝑇. Then there exist at least one edge say 𝑥𝑥
2∈ 𝑇 such that 𝑥𝑥
𝑖∉ 𝑊. Assume that 𝑥𝑥
𝑖= 𝑦𝑧
𝑖for some 𝑖 (1 ≤ 𝑖 ≤ 𝑎 − 2). Then the edge y𝑧
𝑖does not lie on any total edge-to-vertex Steiner 𝑊
𝑒𝑣-tree of 𝐺 and so 𝑊 is not an total edge-to-vertex Steiner set. Now assume that 𝑥𝑥
𝑖= 𝑥
1𝑣
𝑗for some 𝑗 (1 ≤ 𝑗 ≤ 𝑏 − 𝑎 − 4).
Then the edges 𝑥
1𝑣
𝑗does not lie on any total edge -to-vertex Steiner 𝑊
𝑒𝑣-tree of 𝐺 and so 𝑊 is not a minimal edge-to-vertex Steiner set of 𝐺 so that 𝑠
𝑡𝑒𝑣+(𝐺) = b.
G Figure 2.2
