Tree Domination Number Of Middle And Splitting Graphs

: Let G = (V, E) be a connected graph. A subset D of V is called a dominating set of G if N[D] = V. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by  (G). A dominating set D of a graph G is called a tree dominating set (ntr - set) if the induced subgraph  D  is a tree. The tree domination number γ tr (G) of G is the minimum cardinality of a tree dominating set. The Middle Graph M(G) of G is defined as follows. The vertex set of M(G) is V(G)  E(G). Two vertices x. y in the vertex set of M(G) are adjacent in M(G) if one of the following holds. (i) x, y are in E(G) and x, y are adjacent in G. (ii) x  V(G), y  E(G) and y is incident at x in G. Let G be a graph with vertex set V(G) and let V′(G) be a copy of V(G). The splitting graph S(G) of G is the graph, whose vertex set is V(G)  V′(G) and edge set is {uv, u′v and uv′: uv  E(G)}. In this paper we study the concept of tree domination in middle and splitting graphs.


Mathematics Subject Classification
The graph G ο K1 is obtained from the graph G by attaching a pendent edge to all the vertices of G.The total graph T(G) of a graph G is a graph such that the vertex set T(G) corresponds to the vertices and edges of G and two vertices are adjacent in T(G) if and only if their corresponding elements are either adjacent or incident in G.A covering graph is a subgraph which contains either all the vertices or all the edges corresponding to some other graph.A subgraph which contains all the vertices is called a line(edge) covering.A subgraph which contains all the edges is called a vertex covering.The Middle Graph M(G) of G is defined as follows.The vertex set of M(G) is V(G)E(G).Two vertices x. y in the vertex set of M(G) are adjacent in M(G) if one of the following holds.(i) x, y are in E(G) and x, y are adjacent in G.
(ii) xV(G), yE(G) and y is incident at x in G. Let G be a graph with vertex set V(G) and let V′(G) be a copy of V(G).The splitting graph S(G) of G is the graph, whose vertex set is V(G)  V′(G) and edge set is {uv, u′v and uv′: uvE The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by (G).Xuegang Chen, Liang Sun and Alice McRac [9] introduced the concept of tree domination in graphs.A dominating set D of G is called a tree dominating set, if the induced subgraph D is a tree.The minimum cardinality of a tree dominating set of G is called the tree domination number of G and is denoted by tr(G).In this paper we study the concept of tree domination in middle and splitting graphs.
2. PRIOR RESULTS Theorem 2.1: [2] For any graph G, (G)  (G).Theorem 2.2: [9] For any connected graph G with n  3, γtr(G) ≤ n  2. Theorem 2.3: [9] For any connected graph G with γtr(G) = n  2 iff G  Pn (or) Cn.Theorem 2.4: [9] For every support is a member of every tree dominating set of G, γtr(G) = s, where S is the set of support vertices and │S│= s.Theorem 2.5: [9] For every connected graph G with n vertices, γtr(G) = n -2 if and only if G is isomorphic to Pn or Cn.

MAIN RESULTS
In this section, tree domination numbers of middle and splitting graphs are found.

TREE DOMINATION NUMBER IN MIDDLE GRAPHS
The Middle Graph M(G) of G is defined as follows.The vertex set of M(G) is V(G)E(G).Two vertices x. y in the vertex set of M(G) are adjacent in M(G) if one of the following holds. (i) x, y are in E(G) and x, y are adjacent in G. (ii) xV(G), yE(G) and y is incident at x in G.In this section, tree domination numbers for middle graphs of some particular graphs are found and the graphs for which tr(M(G)) = 1, 2 and n -2 are characterized.For any path Pn on n vertices, γtr(M(Pn)) = n -1, n ≥ 3.

Proof:
The set L(Pn) is a minimum tree dominating set of M(Pn), since L(Pn) is isomorphic to Pn-1 and each vertex of G in M(G) is adjacent to atleast one vertex in L(Pn).Therefore, γtr(M(Pn)) = |V(L(Pn)) | = n -1, n ≥ 3.

Proof:
The pendant vertices of Pn ο K1 are pendant vertices of M(Pn ο K1).The supports are the vertices in M(Pn ο K1) corresponding to pendant edges in Pn ο K1.Any dominating set of M(Pn ο K1) contains all these supports.To get a tree dominating set of M(Pn ο K1), vertices corresponding to edges of Pn in Pn ο K1 is to be included.But the subgraph of M(Pn ο K1) induced by this dominating set contains cycles.Therefore, there exists no tree dominating set for M(Pn ο K1) and hence γtr(M(Pn ο K1)) = 0, n ≥ 2. Theorem 3.1.5:γtr(M( n P )) = n -1, where n P is the complement of Pn, n ≥ 5.
In the following, the connected graphs G for which γtr(M(G)) = 1, 2 are characterized.Theorem 3.1.7.
For any connected graph G, γtr(M(G)) = 1 if and only if G  K2.Proof: For any connected graph G on atleast three vertices, γtr(M(G)) = 2 if and only if there exists two adjacent edges e1 and e2 in G such that (i) {e1, e2} is an edge cover of G and (ii) all the edges of G are adjacent to atleast one of e1 and e2.

Proof:
Assume γtr(M(G)) = 2. Let D be a tree dominating set of M(G) such that |D| = 2. Since the subgraph of M(G) induced by vertices of G is totally disconnected, either two vertices of D are vertices of L(G) (or) one vertex is in G and the other vertex is in L(G).

Case 1. Two vertices of D are vertices of L(G)
Let e1, e2D.Then e1, e2 are edges in G. Let e3E(G) be such that e3 is not adjacent to both e1 and e2 in G. Then e3L(G) is not adjacent to any of e1 and e2.Therefore, all the edges are adjacent to atleast one of e1 and e2.
Let u be a vertex of G in M(G).Then u is adjacent to one of e1 and e2 in M(G).Therefore, {e1, e2} is an edge cover of G. Case 2. One vertex is in G and the other is in L(G) Let D = {u, e} be a tree dominating set of M(G), where uV(G) and eV(L(G)).Then eE(G) is incident with u.Let e = (u, v), where vV(G).Let e1 be an edge of G adjacent to e and e1 = (v, w), where wV(G).Then wV(M(G)) is not adjacent to any of u and e.Let e2 = (w, x)E(G) be not adjacent to e (w, xV(G)).Then none of e2, w, x in M(G) is adjacent to any of u and e.Therefore, G  K2.But, γtr(M(K2)) = 1.

Theorem 3.1.9:
Let G be a connected graph with n vertices and m edges.Then γtr(M(G)) = n + m -2 if and only if G is isomorphic to K2.

Proof:
By Theorem 2.5., "For every connected graph G with n vertices, γtr(G) = n -2 if and only if G is isomorphic to Pn or Cn", γtr(M(G)) = n + m -2 if and only if M(G) is isomorphic to Pn+m or Cn+m.If G contains two adjacent edges, then M(G) contains a triangle.If G  2K2, then M(G)  2P3.Therefore, G contains exactly one edge and M(G) is isomorphic to P3.Also, there is no graph G for which M(G) is a cycle.Theorem 3.1.10: Let G be a connected graph on atleast three vertices.Then any tree dominating set D of L(G) is a tree dominating set of M(G) if and only if the set D′ of edges in G corresponding to vertices in D is (i) an edge cover of G (ii) each edge in G is adjacent to atleast one of the edges in D′.

Proof:
Let D be a tree dominating set of L(G) and let D′ be the set of all edges of G corresponding to vertices in D.
Assume conditions (i) and (ii).By (i), D dominates all the vertices of G in M(G).By (ii), D dominates all the vertices of L(G) in M(G).Since D is a tree in M(G), D is also a tree dominating set of M(G).
Conversely, if D′ is not an edge cover of G, then there exists a vertex u in G not incident with any of the edges in D′.Then the vertex u in M(G) is not adjacent to any of the vertices in D. Let e be an edge not adjacent to any of the edges in D′.Then the vertex e in M(G) is not adjacent to any of the vertices in D. Therefore, conditions (i) and (ii) hold.Theorem 3.1.11: Let G be a connected graph on atleast three vertices.Any tree dominating set of M(G) contains atmost two vertices of G.

Proof:
Let D be a tree dominating set of M(G) such that D contains atleast three vertices of G. Let v1, v2, v3 be any three vertices of G in D. Since the subgraph of M(G) induced by {v1, v2, v3} is totally disconnected, D contains vertices of L(G) such that the corresponding edges in G are incident with v1, v2, v3.Since D is a tree in M(G), adjacent vertices in D are not the vertices of G. Let e1 = (v1, v2) and e2 = (v2, v4), where v4V(G).Then e1 and e2 in V(L(G)) are adjacent in M(G) and D contains a cycle and is not a tree.Therefore, D contains atmost two vertices of G.

TREE DOMINATION NUMBER IN SPLITTING GRAPHS
In this section, tree domination numbers of splitting graphs of some standard graphs are obtained.

Definition 3.2.1:
Let G be a graph with vertex set V(G) and let V′(G) be a copy of V(G).The splitting graph S(G) of G is the graph, whose vertex set is V(G)  V′(G) and edge set is {uv, u′v and uv′: uvE(G)}.

Example 3.2.1:
In the graph G given in Figure 2.4, the set {v3, v4} is a minimum tree dominating set of both G and S(G) and γ(G) = γtr(G) = γtr(S(G)) = 2. Observation 3.2.1: For any connected graph G, γtr(G) ≤ γtr(S(G)).This is illustrated by the following examples Example 3.2.2: In the graph G given in Figure 3.4, the set {v3, v4} is a minimum tree dominating set of both G and S(G) and γtr(G) = γtr(S(G)) = 2.

Example 3.2.3:
In the graph G given in Figure 2.7, minimum tree dominating set of G is {v3} and γtr(G) = 1.In the graph S(G), minimum tree dominating set of S(G) is {v1, v3} and γtr(S(G)) = 2.
Any tree dominating set of G containing atleast two vertices is also a tree dominating set of S(G).

Proof:
Let D be a tree dominating set of G. Then D is a tree and each vertex in V(G) -D is adjacent to atleast one vertex in D. Since D  V(S(G)), D is also a tree in S(G).Each vertex of G in V(S(G)) -D is adjacent to atleast one vertex in D. Let vV(G) -D and let v be adjacent to u in D. Then the duplicate vertex v′ of v is also adjacent to u.Since |D| ≥ 2 and D is a tree, u is adjacent to atleast one vertex in D  V(G).Let wD be adjacent to u.Then the duplicate vertex u′ of u is adjacent to w and w′ is adjacent to u.Therefore, each vertex of V′(G) in V(S(G)) -D is adjacent to atleast one vertex in D of S(G) and D is also a tree dominating set of S(G).

Definition 3.2.2: Shadow Graph
Shadow Graph D2(G) of a connected graph G is constructed by taking two copies of G, say G′ and G′′.Join each vertex u′ in G′ to the neighbours of the corresponding vertex u′′ in G′′.

Example 3.2.5:
In the graph G and D2(G) given in Figure 3.6, the set {v2, v3} is a minimum tree dominating set of both G and D2(G) and γtr(G) = γtr(D2(G)) = 2. Theorem 3.2.10: Let G be a connected graph.Any tree dominating set of G containing atleast two vertices is also a tree dominating set of D2(G).

Proof:
Let D be a tree dominating set of G containing atleast two vertices and let G′ and G′′ be two copies of G. Then D is a tree dominating set of G′.Let uG′ be such that uD and u′′G′′, Since D is a tree, u′ is adjacent to a vertex, say v in D. Then u′′ is adjacent to v in D. Therefore, all the vertices in G′′ is adjacent to atleast one vertex in D and hence D is a tree dominating set of D2(G).

: 05C69 1 INTRODUCTION
The graphs considered here are nontrivial, finite and undirected.The order and size of G are denoted by n and m respectively.If D  V, then vD N(D) = N(v)  and N[D] = N(D)  D where N(v) is the set of vertices of G which are adjacent to v. The concept of domination in graphs was introduced by Ore[4].