ISSN 2319-8133 (Online)
(An International Research Journal), www.compmath-journal.org
On Independence Number of Double Vertex Graphs
Roopa Prabhu
*and K. Manjula
1Department of Science,
S. J(Govt). Polytechnic, Sheshadri Road, Bangalore-01, Karnataka, INIDA.
1
Department of Mathematics,
Bangalore Institute of Technology, Bangalore, Karnataka, INDIA.
email:roopa_prabhu@rediffmail.com
(Received on: August 22, 2018) ABSTRACT
A set of vertices in a graph 𝐺 is independent if no two vertices in the set are adjacent. The maximum of the cardinalities of all vertex independent sets of G is vertex independence number of G denoted by 𝛽(G). In this paper we obtain some results on vertex independence number of double vertex graphs and also determine vertex independence number of double vertex graph of complete graph, bipartite graph and star graph.
2010 Mathematics Subject Classification: 05C69.
Keywords: Double vertex graph, vertex independence number, edge independence number.
1. INTRODUCTION
All graphs considered are simple, undirected and finite. Any undefined terminologies and notations we refer to
1. There are many graph functions with which one can construct a new graph from a given graph or set of graphs. One such graph function is double vertex graph introduced by Alavi et al.
2. Let 𝐺 = (𝑉, 𝐸) be a graph of order 𝑛 ≥ 2 then the double vertex graph of 𝐺 denoted by 𝑈
2(𝐺) is the graph whose vertex set consists of all ( 𝑛
2) unordered pairs from 𝑉 such that two vertices {𝑥, 𝑦} and {𝑢, 𝑣} are adjacent if and only if
|{𝑥, 𝑦} ∩ {𝑢, 𝑣}| = 1 and if 𝑥 = 𝑢, then 𝑦 and 𝑣 are adjacent in 𝐺. We describe the vertex set
of 𝑈
2(𝐺) as 𝑉(𝑈
2(𝐺)) = {{𝑣
𝑖, 𝑣
𝑗}/𝑣
𝑖, 𝑣
𝑗∈ 𝑉(𝐺); 1 ≤ 𝑖 ≤ 𝑛 − 1; 𝑖 + 1 ≤ 𝑗 ≤ 𝑛}. If 𝐺 has 𝑛
vertices and 𝑚 edges, then 𝑈
2(𝐺) has 𝑛(𝑛 − 1)/2 vertices and 𝑚(𝑛 − 2) edges. For each
edge of 𝐺 there are 𝑛 − 2 edges of 𝑈
2(𝐺). A vertex {𝑥, 𝑦} ∈ 𝑉(𝑈
2(𝐺)) is a line pair if and only if 𝑥𝑦 ∈ 𝐸(𝐺) otherwise a non line pair
3. For more results on double vertex graphs we refer
2 and 3.
A set 𝑆 of vertices of a graph 𝐺 is independent if no two vertices in 𝑆 are adjacent.
The vertex independence number 𝛽(𝐺) of a graph 𝐺 is maximum of the cardinalities of all the vertex independent sets of 𝐺. A subset 𝑀 of the edge set of 𝐺 is called matching in 𝐺 if no two of the edges in 𝑀 are adjacent. The edge independence number 𝛽
1(G) of 𝐺 is the cardinality of maximum edge independent subset of 𝐺.
The degree of a vertex 𝑣 in a graph 𝐺 is the number of edges of 𝐺 incident with 𝑣 denoted by 𝑑𝑒𝑔𝑣. The maximum (minimum) degree among the vertices of 𝐺 is denoted by
∆(𝐺)(𝛿(𝐺)). A vertex of degree zero is isolated vertex and a vertex of degree one is called a pendant vertex. A vertex adjacent to a pendant vertex is called as non pendant vertex. A neighborhood of a vertex 𝑣 denoted by 𝑁(𝑣) is the set of vertices adjacent to 𝑣. A closed neighborhood of vertex 𝑣 is {𝑣} ∪ 𝑁(𝑣). For a vertex 𝑢 in a connected graph 𝐺 the eccentricity 𝑒(𝑢) of 𝑢 is the distance between 𝑢 and a vertex farthest from 𝑢 in 𝐺. The minimum eccentricity among the vertices of 𝐺 is its radius 𝑟(𝐺) and the maximum eccentricity is its diameter 𝑑𝑖𝑎𝑚(𝐺). Two graphs 𝐺 and 𝐻 are said to be isomorphic if there exists an one to one correspondence 𝛷 from 𝑉(𝐺) to 𝑉(𝐻) such that every two adjacent vertices of 𝐺 are mapped to adjacent vertices of 𝐻 and every two non adjacent vertices of 𝐺 is mapped to non adjacent vertices of 𝐻 denoted by 𝐺 ≅ 𝐻. A graph 𝐺 is a bipartite graph if 𝑉(𝐺) can be partitioned into two subsets 𝑈 and 𝑊 called partite sets such that every edge of 𝐺 joins a vertex of 𝑈 and a vertex of 𝑊. If every vertex of 𝑈 joins every vertex of 𝑊 then 𝐺 is a complete bipartite graph 𝐾
𝑚,𝑛where |𝑈| = 𝑚 and |𝑊| = 𝑛. As usual 𝑃
𝑛,𝐶
𝑛and 𝐾
𝑛are respectively the path, cycle, complete graph of order 𝑛. The vertex independence number of Product Graphs are well studied in
5,6 and 7. For all the undefined terminologies we refer to
1.
2. MAIN RESULTS
Definition 2.1[2]: Let 𝐺 = (𝑉, 𝐸) be a graph of order 𝑛 ≥ 2 then the double vertex graph of 𝐺 denoted by 𝑈
2(G) is the graph whose vertex set consists of all ( 𝑛
2) unordered pairs from 𝑉 such that two vertices {𝑥, 𝑦} and {𝑢, 𝑣} are adjacent if and only if |{𝑥, 𝑦} ∩ {𝑢, 𝑣}| = 1 and if 𝑥 = 𝑢, then 𝑦 and 𝑣 are adjacent in 𝐺. Fig 1 shows the double vertex graph of cycle 𝐶
5where 𝑉(𝐶
5) = {1,2,3,4,5}.
Remark 2.2[2]: For a connected graph 𝐺, 𝑈
2(𝐺) is a complete graph if and only if 𝐺 ≅ 𝐾
3. Remark 2.3: For any independent set 𝑆 of 𝑈
2(𝐺), {𝑥, 𝑦} ∪ 𝑆 is also independent if and only if one of the following condition holds.
i) {x, y} ≠ {a, b} for every {a, b} ∈ S
ii) if x = a then y ∉ N[b] or y = b then x ∉ N[a]
Remark 2.4: For a graph 𝐺, 1 ≤ 𝛽(𝑈
2(𝐺)) ≤ ( 𝑛
2).
Theorem 2.5: For a connected graph 𝐺, 𝛽(𝑈
2(𝐺)) = 1 if and only if 𝐺 ≅ 𝐾
3.
Fig 1: Double vertex graph of C5
Proof: For 𝐺 ≅ 𝐾
3, 𝑈
2(𝐾
3) ≅ 𝐾
3then the maximum vertex independent set of 𝑈
2(𝐾
3) is a singleton set hence 𝛽(𝑈
2(𝐺)) = 1. Conversely, 𝛽(𝑈
2(𝐺)) = 1 implies maximum vertex indep-endent set 𝑆 of 𝑈
2(𝐺) has a full degree vertex. The only possibilities are 𝐺 ≅ 𝐾
3or 𝑃
3. For 𝐺 ≅ 𝑃
3, the maximum vertex independent set 𝑆 of 𝑈
2(𝑃
3) has two elements. Therefore 𝐺 ≅ 𝐾
3.
Theorem 2.6: For a connected graph 𝐺, 𝛽(𝐺) ≤ 𝛽(𝑈
2(𝐺)).
Proof: Let 𝑆
′= {{𝑣
𝑖}: 1 ≤ 𝑖 ≤ 𝛽(𝐺)} and 𝑆 be the maximal vertex independent sets of 𝐺 and 𝑈
2(𝐺) respectively such that |𝑆
′| = 𝛽(𝐺) and |𝑆| = 𝛽(U
2(G)). Construct a set 𝑋 = {{𝑣
𝑖, 𝑣
𝑗}\ 𝑣
𝑖, 𝑣
𝑗∈ 𝑆
′; 1 ≤ 𝑖 ≤ 𝛽(𝐺) − 1, 𝑖 + 1 ≤ 𝑗 ≤ 𝛽(𝐺), 𝑗 ≠ 𝑖}. Clearly 𝑋 is an independent subset of 𝑉(𝑈
2(𝐺)) as each 𝑣
𝑖and 𝑣
𝑗are independent and |𝑋| = (
𝛽(𝐺)2). The
|𝑋| ≤ |𝑆| as 𝑆 is maximal. Hence |𝑆
′| ≤ |𝑋| ≤ |𝑆| which implies 𝛽(𝐺) ≤ 𝛽(𝑈
2(𝐺)).
Equality is attained for 𝐺 ≅ 𝐾
1,3, 𝐾
3, 𝑃
3, triangle with a tail attached to its one vertex.
Theorem 2.7: For a connected graph 𝐺, ⌊
𝑛2⌋ ≤ 𝛽(𝑈
2(𝐺)).
Proof: Let 𝑆 be a maximum vertex independent set of 𝑈
2(𝐺) such that |𝑆| = 𝛽(U
2(G)). We prove the theorem for odd and even 𝑛 seperately.
Case (i): When 𝒏 is odd,
Consider a proper subset 𝑋 = {{𝑣
𝑖, 𝑣
𝑗}: 𝑖 = 1,3,5, … . . , 𝑛 − 2; 𝑗 = 𝑖 + 1} of 𝑉(𝑈
2(𝐺)).
Clearly 𝑋 is an independent set as no two elements of 𝑋 have a common element and < 𝑋 >
a null graph, also |𝑋| = ⌊
𝑛2⌋. Therefore |𝑋| ≤ |𝑆| as 𝑆 is a maximum independent set. Hence
⌊
𝑛2
⌋ ≤ 𝛽(U
2(G)) (1) Case (ii): When 𝒏 is even,
In this case 𝑌 = {{𝑣
𝑖, 𝑣
𝑗}: 𝑖 = 1,3,5, … . . . . , 𝑛 − 1; 𝑗 = 𝑖 + 1} is a proper subset of 𝑉(𝑈
2(𝐺)) and is independent as no two vertices of 𝑌 have common element. Clearly |𝑌| =
𝑛2and |𝑌| ≤
|𝑆|. Hence
C
5𝑛
2 ≤ 𝛽(U
2(G)) (2) From (1) and (2) we have ⌊
𝑛2⌋ ≤ 𝛽(U
2(G)).
Theorem 2.8: For a connected graph 𝐺 with no induced subgraph isomorphic to 𝐶
3, 𝑑𝑖𝑎𝑚(𝐺) ≤ 2 ,then vertices of U
2(G) corresponding to the line pairs of 𝐺 form a maximal vertex independent set of U
2(G).
Proof: 𝐺 = (𝑉, 𝐸) be a connected graph of order 𝑛 and size 𝑚. Let 𝑑𝑖𝑎𝑚(𝐺) ≤ 2, Partition the vertices of U
2(G) into two disjoint sets 𝑋 and 𝑌 where 𝑋 corresponds to line pairs and 𝑌 corresponds to non line pairs of 𝐺 respectively. Since 𝐺 has no induced subgraph isomorphic to 𝐶
3, no two elements of 𝑋 are adjacent as two line pairs in 𝑈
2(𝐺) are adjacent if and only if the corresponding edges lie in a triangle
3. Therefore 𝑋 is independent and also maximal as every element in 𝑌 is adjacent to atleast two elements in 𝑋. Hence 𝑋 ∪ {𝑢, 𝑣} for any {𝑢, 𝑣} ∈ 𝑉(𝑈
2(𝐺)) − 𝑋 is not independent. Therefore 𝑋 is maximal.
Theorem 2.9: For a connected graph 𝐺 with no induced subgraph isomorphic to 𝐶
3and 𝑑𝑖𝑎𝑚(𝐺) ≥ 3 then 𝛽(U
2(G)) > 𝑚.
Proof: 𝐺 = (𝑉, 𝐸) be a connected graph of order 𝑛 and size 𝑚. Partition 𝑉(𝑈
2(𝐺)) into two disjoint sets 𝑋 and 𝑌 where 𝑋 corresponds to line pairs and 𝑌 corresponds to non line pairs of 𝐺 respectively. Since 𝐺 has no induced subgraph isomorphic to 𝐶
3, 𝑋 is an independent set[3].
Let 𝑃
𝑘: 𝑣
1− 𝑣
2− 𝑣
3− ⋯ … . . . −𝑣
𝑘+1be a path of length 𝑘 ≥ 3 in 𝐺 then the vertex {𝑣
1, 𝑣
𝑘+1} corresponding to the endvertices of 𝑃
𝑘is a non line pair in 𝑈
2(𝐺) which is not adjacent to any of the vertices in 𝑋. Hence 𝑋 ∪ {𝑣
1, 𝑣
𝑘+1} is an independent set in 𝑈
2(𝐺).
Therefore for any maximum vertex independent set of U
2(G), 𝛽(U
2(G)) is atleast |𝑋| + 1 which implies 𝛽(𝑈
2(𝐺)) > 𝑚.
Theorem 2.10: For a connected graph 𝐺, 𝛽
1(𝐺) ≤ 𝛽(𝑈
2(𝐺)). Equality holds for 𝐺 ≅ 𝐾
𝑛, 𝐾
4− 𝑒,triangle with a tail attached to its one vertex.
Proof: Let 𝑉(𝑈
2(𝐺)) = 𝑋 ∪ 𝑌 where 𝑋 corresponds to line pairs and 𝑌 corresponds to non line pairs of 𝐺 respectively. Let 𝑆
′be a maximum edge independent set of 𝐺 such that |𝑆
′| = 𝛽
1(𝐺) the edge independence number. Then 𝑆
′⊆ 𝑋 and for any maximum independent set 𝑆 of 𝑈
2(𝐺), |𝑆
′| ≤ |𝑆|. Hence 𝛽
1(𝐺) ≤ 𝛽(U
2(G)).
For 𝐺 ≅ 𝐾
𝑛, |𝑆
′| = |𝑆| as the maximum vertex independent set of 𝑈
2(𝐾
𝑛) corresponds to the maximum independent edge set of 𝐾
𝑛.
Theorem 2.11: For a complete graph 𝐾
𝑛, 𝛽(𝑈
2(𝐾
𝑛)) = ⌊
𝑛2⌋.
Proof: 𝑈
2(𝐾
𝑛) is the line graph of 𝐾
𝑛2. The independence number of line graph of a graph is the matching number of the graph
4and matching number of the 𝐾
𝑛is ⌊
𝑛2⌋.
Hence 𝛽(𝑈
2(𝐾
𝑛)) = ⌊
𝑛2⌋.
Remark 2.12: For any connected graph 𝐺 with 𝑞 ≥ 2 pendant vertices, ( 𝑞
2) ≤ 𝛽(𝑈
2(𝐺)).
Theorem 2.13: For a tree 𝑇, (
𝑞2) + (𝑚 − 𝑞) ≤ 𝛽(𝑈
2(𝑇)) where 𝑞 is the number of pendant vertices of 𝑇.
Proof: Let 𝑆 be a maximum vertex independent set of 𝑈
2(𝑇) such that |𝑆| = 𝛽(U
2(T)).
Without loss of generality let 𝑋 = {{𝑣
𝑖}: 1 ≤ 𝑖 ≤ 𝑞 } be a set of pendant vertices of 𝑇. Clearly 𝑋 is an independent set. Now the set 𝑌 = {{𝑣
ℎ, 𝑣
𝑘}: 1 ≤ ℎ ≤ 𝑞 − 1; ℎ + 1 ≤ 𝑘 ≤ 𝑞} whose elements are non line pairs of 𝑇 is a proper subset of 𝑉(𝑈
2(𝑇)) and |𝑌| = ( 𝑞
2). Then 𝑌 is an independent set as no two elements of 𝑌 are adjacent. Let 𝑚 − 𝑞 non pendant edges 𝑇 corresponds to a subset 𝑍 of 𝑉(𝑈
2(𝑇)). Then 𝑍 is independent so is 𝑌 ∪ 𝑍 which implies |𝑌 ∪ 𝑍| ≤ |𝑆| as 𝑆 is maximum. Hence |𝑌| + |𝑍| ≤ 𝛽(U
2(T)) implies (
𝑞2) + (𝑚 − 𝑞) ≤ 𝛽(U
2(T)).
Theorem 2.14: For a bipartite graph 𝐺, 𝑛~𝑚 = 𝑑 where 𝑚,𝑛 and 𝑑 are positive integers 𝛽(𝑈
2(𝐺)) = { (
𝑚2) + (
𝑛2) 𝑤ℎ𝑒𝑛 𝑚𝑖𝑛{𝑚, 𝑛} ≤ (𝑑
2 ) 𝑚𝑛 𝑤ℎ𝑒𝑛 𝑚𝑖𝑛{𝑚, 𝑛} > (𝑑
2 )
Proof: Let 𝐺 be a bipartite graph with partite sets 𝑈 = {𝑣
1, 𝑣
2, 𝑣
3, 𝑣
4, 𝑣
5, … . , 𝑣
𝑚} and 𝑊 = {𝑢
1, 𝑢
2, 𝑢
3, … 𝑢
𝑛}. 𝑉(𝑈
2(𝐺)) can be partitioned into two disjoint sets 𝑉
1and 𝑉
2where 𝑉
1= {{𝑣
𝑖, 𝑣
𝑗} ∪ {𝑢
𝑟, 𝑢
𝑡}\1 ≤ 𝑖 ≤ 𝑚 − 1, 𝑖 + 1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑟 ≤ 𝑛 − 1, 𝑟 + 1 ≤ 𝑡 ≤ 𝑛} and
|𝑉
1| = ( 𝑚 2 ) + (
𝑛
2). Further 𝑉
2consisting of 𝑚𝑛 elements can be partitioned into two disjoint sets 𝑃 and 𝑄 such that elements of 𝑃 corresponds to the line pairs of 𝐺 and 𝑄 = 𝑉
2− 𝑃. Then
|𝑃| = |𝐸(𝐺)| and |𝑄| = 𝑚𝑛 − |𝐸(𝐺)| and both are independent. Therefore both 𝑉
1and 𝑉
2are independent subsets of 𝑉(𝑈
2(𝐺)). The maximality of 𝑉
1and 𝑉
2can be checked as below.
Case (i) 𝑚𝑖𝑛{𝑚, 𝑛} = 𝑚 then 𝑛~𝑚 = 𝑑 implies 𝑛 = 𝑚 + 𝑑;
Then 𝑚𝑛 = 𝑚(𝑚 + 𝑑) = 𝑚
2+ 𝑚𝑑 = 𝑚
2+ (𝑑 − 1)𝑚 + 𝑚 (3)
(
𝑚2) + (
𝑛2) =
𝑚(𝑚−1)2
+
(𝑚+𝑑)(𝑚+𝑑−1)2
=
𝑚2−𝑚+𝑚2+2𝑑𝑚−𝑚+𝑑2−𝑑2