The Total Strong Split Domination Number of Graphs

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 5 Issue 2 || February. 2017 || PP-01-03
www.ijmsi.org 1 | Page
The Total Strong Split Domination Number of Graphs
T.Nicholas, T.Sheeba Helen
Department of Mathematics, St.Judes College, Thoothoor, TamilNadu, India.
Department of Mathematics, Holy Cross College (Autonomous), Nagercoil, TamilNadu, India.
ABSTRACT: A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced
subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number
γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total
strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of
γtss(G) are established.
KEYWORDS: Domination number, split domination number, total domination number, strong split domination
number, total strong split domination number.
Subject Classification: 05C
I. INTRODUCTION
The graphs considered here are finite, undirected, without loops, multiple edges and have atmost one component
which is not complete. For all graph theoretic terminology not defined here, the reader is referred to [2]. A set of
vertices D in a graph G is a dominating set, if every vertex in V-D is adjacent to some vertex in D. The
domination number γ(G) is the minimum cardinality of a dominating set. This is well studied parameter as we
can see from [3]. For a comprehensive introduction to theoretical and applied facts of domination in graphs the
reader is directed to the book [3]. A dominating set D of a graph G is a total dominating set if the induced sub
graph < D > has no isolated vertices. The total domination number γt(G) is the minimum cardinality of a total
dominating set of G.This concept was introduced by Cockayne, Dawes and Hedetniemi in [1].
V. R. Kulli and B. Janakiram introduced the concept of split domination in [6]. A dominating set D of a graph G
= (V, E) is a split dominating set if the induced subgraph < V-D > is disconnected. The split domination number
γs(G) is the minimum cardinality of a split dominating set.
Strong split domination was introduced by V. R. Kulli and B. Janakiram in [7]. A dominating set D of a graph G
= (V, E) is a strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two
vertices. The strong split domination number γss(G) is the minimum cardinality of a strong split dominating set.
A total dominating set D of a connected graph G is a total split dominating set if the induced subgraph < V-D >
is disconnected. The total split domination number γts(G) is the minimum cardinality of a strong split
dominating set. This concept was introduced by B. Janakiram, Soner and Chaluvaraju in [5].
We introduce a new concept namely total strong split domination number. A total dominating set D of a
connected graph G is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected
with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total
strong split dominating set.
II. RESULTS
Theorem 1. For any graph G , γs(G) γss(G) γtss(G)
Proof: This follows from the fact that every total strong split dominating set of G is a strong split dominating set
of G and every strong split dominating set of G is a split dominating set of G.
The following two characterizations are easy to see, hence we omit their proofs.
Theorem 2. A total dominating set D of a graph G is a total strong split dominating set if and only if the
following conditions are satisfied.
(i) V-D has atleast two vertices
(ii) For any vertices u, v V-D, every u, v path contains atleast one vertex of D
Theorem 3. A total strong split dominating set of G is minimal if for each vertex v
there exists a vertex u V-D such that u is adjacent to v.
We now consider a lower bound on γtss(G) in terms of the minimum degree, the order and the size of G.
Theorem 4. If G has no isolated vertices and p 3, then γtss(G),
where is the minimum degree of G.

Proof:
Let D be a γtss-set of G.Then every vertex in V-D is adjacent with atleast vertices in D. This implies that q
Thus the theorem follows from the fact that
Theorem 5. For any connected graph G, γtss(G) = , where is the vertex covering number.
Proof: Let S be the minimum independent set of vertices in G. Then V-S is the total strong split dominating set
of G. v V-S, then there exists a vertex u S such that v is adjacent to u.Then by Theorem 3, V-S is minimal
and is the minimal covering for G. = . This proves the theorem.
Corollary 5.1. For any graph G, γtss(G) q
Proof: By (3) γtss(G) =
= p -
Where is the independence number of G.
Thus the result follows from the fact that γ(G) p – q.
Now we list the exact values of γtss(G) for some standard graphs.
Proposition 6.1 For any cycle Cn with n 6 vertices
γtss(Cn) =
Proof: Let V(Cn) = {v0, v1, v2, v3, . . . , vn-1} be the vertex set of the cycle Cn. Let D be the total strong split
dominating set of Cn. Consider the sets,
D1 = { v3i, v3i+1 / i = 0, 1, 2, . . . , } when .
D2 = { v3i, v3i+1 / i = 0, 1, 2, . . . , } { vn-2 } when .
D3 = { v3i, v3i+1 / i = 0, 1, 2, . . . , } { vn-3, vn-2 } when .
The above three sets achieve the total strong split property of Cn in the respective parity conditions.
Corollary 6.2 Let G be a Hamiltonian graph on n vertices. Then γtss(G) γtss(Cn)
Proposition 6.3 For any path Pn with n 4 vertices
γtss(Pn) =
Proof: Let Pn be the path of order n. V(Pn) = { v0, v1, v2, v3, . . . , vn-1}. Let D be the total strong split dominating
set of Pn. When n = 4, V(P4) = { v0, v1, v2, v3 }.Then D = { v1, v2 }.
When n = 5, V(P5) = { v0, v1, v2, v3, v4 }.Then D = { v1, v2, v3 }.
If n 0 (mod 3) then D contains v3i+1, v3i+2 where i = 0, 1, . . . , .
D = { v1, v2, v4, v5, . . . , vn-2, vn-1 }. Hence 2 =
If n 1 (mod 3) then D contains v3i+1, v3i+2 where i = 0, 1, . . . , .
D = { v1, v2, v4, v5, . . . , vn-3, vn-2 }. Thus 2 =
If n 2 (mod 3) then D has v3i+1, v3i+2 where i = 0, 1, . . . , and also vn-2.
D = { v1, v2, v4, v5, . . . , vn-4, vn-3, vn-2 }. In this case 2 +1 =
Hence the result follows.
Proposition 6.4 For any wheel Wn with n 6 vertices, γtss(Wn) = γtss(Cn) + 1
Proof: Let V(Wn) = {v0, v1, v2, v3, . . . , vn-1} be the vertex set of the wheel Wn. Any minimal total strong split
dominating set must contain the apex vertex v0 since otherwise the induced graph
< V- D > will have an edge v0vi for some vi D which is a contradiction to the strong split condition. Hence D =
D1 {v0} where D1 is any minimal total strong split dominating set.
Proposition 6.5 For any complete bipartite graph Km,n, where 2 m < n, γtss( Km,n ) = m+1

Proof: Let V1 and V2 be the partite sets of the complete bipartite graph Km,n, where 2 m < n and =m
=n. Let D be the minimal total strong split set of Km,n. Since < V- D > is totally disconnected D must
contain all the vertices of one partite set. As D is minimal D contains every vertex of V1. Since < V1 > contains
isolated vertices, D must have any one vertex from the partition V2 so that < V1 {v} > does not contain any
isolated vertex for any v V2. Hence D = < V1 {v} > where v V2 is the minimal total strong split
dominating set of Km,n.
Corollary 6.6 For any Star K1, n with n 4 vertices, γtss(K1, n) = 2.
The double star Sm,n is the graph obtained from the joining centers of K1, m and K1, n with an edge. The centers
of K1, m and K1, n are called central vertices of Sm,n
Proposition 6.7 If Sm,n, 2 is double star, then γtss(Sm,n) = 2
Proof: Let Sm,n be a double star where 1 . Then Sm,n has a total strong split dominating set D
containing the central vertices of Sm,n. Thus the induced subgraph < V- D > contains m+n isolated vertices.
Thus < V- D > is totally disconnected and < D > has no isolated vertices. Hence the theorem holds.
The crown graph Cp K1 is the graph obtained from cycle Cp by attaching a pendant edge to each vertex of the
cycle.
Proposition 6.8 γtss(Cp K1) = p, where p is the length of the cycle Cp.
Proof: Let G = Cp K1. Let D be the minimal total strong split dominating set.
V(G) = { v0, v1, v2, . . . , vn -1} { u0, u1, u2, . . . , un -1 }.
E(G) = { vi vi+1 / i = 0, 1, 2, . . . , n-1, subscript modulo n} { uivi / i = 0, 1, 2, . . . , n-1}. D must have all the
cycle vertices, for otherwise < V-D > does not become totally disconnected. Hence D serves as a total
dominating set, which preserves the strong split prpperty. Infact, D is the unique γtss – set of G.
Theorem 7. For a tree T K1, n, γtss(T) n – ,
where denotes the pendant vertices of T.
Proof: Let T K1, n be a tree with n vertices. Let L be the collection of all pendent vertices.
D be the minimal total strong split dominating set of T. D must contain all the support vertices of T. Let S be the
set of support vertices. If V(T) = S L, then D = S and γtss(T) = n – .
If V(T) S L then there exists a vertex x V(T) such that x S L. In this case D must contain at least one
such vertex x .Hence . In other words, γtss(T) n –
Observation:
(i) For a tree T K1, n vertices having degree equal to one does not belong to γtss –set.
(ii) In a tree vertices adjacent to pendant vertices belong to γtss –set.
(iii) Let T K1 contain only leaf and support vertices, then γtss(T) = where denotes the number of
support vertices of T.
Theorem 8. If H is a connected spanning subgraph of G, then γtss(G) γtss(H)
Proof: H be a connected spanning subgraph of a connected graph G. Let be the dominating set of H. W. l. g
let be a total strong split dominating set of G also.
If every edge in G is incident with any vertex of in G then becomes the total strong split dominating set for
G. If there exists an edge in G which is not incident with an vertex of , then there exists D in G such that
D is a total strong split dominating set for G.
Theorem 9. If G is a connected graph with n , then γtss(G) γc(G)
Proof: Let D be a γtss –set of G. Let be the γc –set of G. By definition < > is connected.
The induced subgraph < V- > contains isolated vertices and connected components. Let
ui . N(ui) either belongs to D or V-D. Hence . Therefore γtss(G) γc(G).
Corollary 10: If G is a connected graph with n vertices then γtss(G) n - where is the
maximum number of pendant vertices in any spanning tree.
Proof: Let G be a connected graph .From [4] we know that γc(G) n – for n By the above theorem
we know that γtss(G) γc(G).Therefore γtss(G) n –
Theorem 11. For any connected graph G with n > 6 vertices, γtss(G) n – 2
Proof: Let G be a connected graph with n > 6 vertices. Let D be a γtss –set of G.

Choose a spanning tree T of G such that T contains minimum number of end vertices. Since every tree contains
at least two pendant vertices u and v. The vertices u,v D.
Hence γtss(G) n – 2.
REFERENCES
[1]. C.J Cockayne, R.M. Dawes and S.T. Hedetniemi, Total Domination in Graphs, Networks, 10 (1980) 211-219
[2]. Harary.F, Graph Theory, Addison-Wesley, Reading, MA, 1972.
[3]. T.W.Haynes, S.T.Hedetneimi, P.J.Slater, Fundamentals of Domination in Graphs, MarcelDekker, New York, 1988.
[4]. S.T. Hedetniemi and R.C. Laskar, Connected domination in graphs, In B. Bollobas,editor, Graph Theory and Combinatorics,
Academic Press, London (1984) 209-218.
[5]. B.Janakiraman,N.D.Soner and B.Chaluvaraju, Total Split Domination in Graphs, Far East J.Appl. Math. 6(2002) 89-95
[6]. V. R. Kulli and B. Janakiram, The split domination number of a graph, Graph Theory Notes of New York. New York Academy of
Sciences, XXXII, 16-19 (1997)
[7]. V. R. Kulli and B. Janakiram, The strong split domination number of a graph, Acta Ciencia Indica, Vol.XXXII M, No. 2 (2006)
715-720
[8]. E.Sampathkumar and H.B.Walikar, The connected domination number of a graph, J.Maths. Phys. Sci., 13 (1979) 607-613.

The Total Strong Split Domination Number of Graphs

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