Triple Connected Domination In Directed Graphs
Vijayalakshmi.B
1and R.Poovazhaki
21
Department of Mathematics, Mount Carmel College, Bangalore [email protected]
2
E.M.G.Yadava womens College, Madurai [email protected]
Abstract
The concept of connectedness plays an important role in many networks. Di- graphs are considered as an excellent modeling tool and are used to model many types of relations amongst any physical situations.In this paper the concept of triple connected domination in directed graph has been introduced by considering the ex- istence of directed path containing three vertices of S. A subset S of V of a digraph Dis said to be triple connected dominating set if S is a dominating set and induced sub digraph < S > is triple connected. The minimum cardinality of triple connected domination number and it is denoted by γtc(D).
AMS Subject Classification:O5C20
Key Words and Phrases: Triple connected digraphs, triple connected domina- tion,tournament,transitive
1 Introduction
Triple connected domination number of graph was introduced by G.Mahadevan [5] and made some bounds for general graphs. In this paper, we have introduced triple connected domination in directed graph.The basic definitions has been referred Arumugam.S[1], G.Chartrand et.al[2],G.Chartrand et.al[3],Jorgen [4].For basic results we have referred Vijayalakshmi.B[7, 8, 9, 10, 11] .
Throughout this paper D = (V, A) is a finite directed graph with neither loops nor multiple arcs (but pairs of arcs are allowed) and G = (V, E) is a undirected graph with neither loops nor multiple edges.
Let D = (V, A) be a digraph.For any vertex u ∈ V , the sets O(u) = {v (u, v) ∈ A}
and I(u) = {v/(v, u) ∈ A} are called outset and inset of u.The in degree and out degree of u are defined by id(u) = |I(u)| and od(u) = |O(u)| .The minimum in degree , the
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minimum out degree, the maximum in degree and maximum out degree of D are denoted by δ−, δ+, ∆−, ∆+ respectively.[1]
An oriented graph is a digraph with no cycle of length two. A tournament is an oriented graph where every pair of distinct vertices are adjacent.[4]
A digraph D is transitive if,for every pair of arcs xy and yz in D such that , x 6= z the arc xz is also in D.[4]
A digraph D is quasi-transitive if, for every triple of distinct vertices of D such that xy and yz are the arcs of D ,there is atleast one arc between x and z . Clearly ,a semi complete digraph is quasi-transitive.[4]
The transitive closure TC(D) of a digraph D is a digraph with V(TC(D))=V(D) and for distinct vertices u,v, the arc uv ∈ A(T C(D)) if and only if D has a (u,v)-path. Clearly if D is strong then TC(D) is complete digraph.[4]
A digraph D on n vertices is round if we can label its vertices v1, v2, ...vn so that each i, we have N+(vi) = {vi+1, ...vi+d+(vi)} and N−(vi) ={vi−d−(vi), ...vi−1}
(all subscripts are taken modulo n). clearly every strong round digraph D is Hamiltonian, since v1, v2, ...vn, v1form a Hamiltonian cycle,whenever v1, v2, ...vnis round labelling.[4]
If D is strong and S is a subset of V(D) such that D-S is not strong,then S is a seperating set.[4]
Directed wheel −C→n+ K1 is the digraph obtained by adding an arc from each vertex from
−→
Cnto the vertex of K1.
Directed star −−−→K1, n is the digraph −−−→K1, n = (V, A) where V = v0, v1, ....vn and A = (v0, v1), (v0, v2)...(v0, vn).The vertex v0 is called the central vertex of the directed star.
The Directed comet,−−→Cm,n where m and n are the positive integers denotes the out-tree obtained by identifying the central vertex of the directed star −−→K1,n with the vertex of di- rected path −p→mof outdegree 0.
A directed wounded spider −−→Sm,nof order m+n+1 is the digraph obtained by subdividing n (1≤ n < m)arcs of the directed star−−→K1,m
If m = n then the digraph −−→Sn,nis called directed spider.
Throughout this paper connected digraph has been taken and out-domination of its anal- ysed .
2 γtc+- for directed paths, directed cycles and complete orientation graph
2.1 Notation
1. γtc+ - set of all vertices in triple connected dominating set S+in D.
Definition 2.1. A subset S+ of V of a digraph D is said to be triple connected dominating set if S+ is a dominating set and induced sub digraph < S+ > is triple connected. The minimum cardinality of triple connected domination number and it is denoted by γtc+(D).
Theorem 2.1. Let D be a directed path.Then γtc+(D) = n− 1, n ≥ 4
Proof. Let D be a directed path and S+be a triple connected dominating set.If D has δ−(D) = δ+(D) = 0then S+ contains all the vertices except the vertex δ+(D) = 0and
the sub digraph < S+ >is triple connected.
∴ γtc+(D) = n− 1.
Theorem 2.2. Let D be a directed cycle .Then γtc+(D) = n− 1,n ≥ 4
Proof. Let D be a directed cycle with δ+(D) = δ−(D) = 1. Let S+ be the triple connected dominating set with n − 1 vertices to make the sub digraph < S+ >is triple connected.
If V − S+has 2 or more vertices then < S+>is not triple connected.
∴ γtc+(D) = n− 1.
Theorem 2.3. Let D be an orientation graph.Then γtc+(D)≤ n − 1.
Proof. Let D be an orientation graph.Then every pair of vertices has an arc and also we know that orientation graph has a spanning path. Let S+ be a triple connected domi- nating set which has atleast 3 vertices and the sub digraph < S+ >is triple connected.
∴ γtc+(D)≤ n − 1.
Theorem 2.4. Let D be a directed star.Then γtc+(D) = 0.
Proof. Let D be a directed star, Case (i): 4+(D) = P and 4−(D) = 1
Let S+ be a dominating set and it has only one vertex which has maximum out degree that dominates all the vertices in V − S+.
∴ < S+>is not triple connected.
Case (ii): 4−(D) = P and 4+(D) = 1
Let S+be a dominating set which has p vertices dominate exactly one vertex which is in V − S+. But S+contains the vertices which are non-adjacent.
∴ < S+>is not connected.
Hence by case (i) and (ii) γtc+(D) = 0.
3 Triple connected domination in various digraphs
Theorem 3.1. Let D be a transitive tournament.Then γtc+(D)≤ n − 1, n ≥ 4
Proof. Since the tournament D is transitive whenever (u, v) and (v, w) are arcs of D then (u, w) is also an arc of D. since the tournament contains a spanning path.
∴ γtc+(D)≤ n − 1.
Theorem 3.2. Let T be a strong tournament. For every integer 4 ≤ k ≤ n there exists a k- cycle through x in T .Then γ+tc(T )≤ n − 1, n ≥ 4.
Proof. Let x, x1, x2, ...xn−1be n vertices. Consider x1, x2...xn−1and T has a n-cycle through x.
Hence x, x1, x2....xn−1, xforms n-cycle . Thus the strong tournament has spanning path.
∴ γtc+(T )≤ n − 1
Theorem 3.3. Let A and B be two distinct strong components of a quasi-transitive digraph D with atleast one arc from A to B. Then γtc+(D)≤ n − 2
Proof. Let A and B be two distinct strong components with an arc from A to B. Then by the choice of x ∈ A and y ∈ B there exists a path from x to y in D. since x does not dominates y, either y −→ x or there exists vertices u, v ∈ V (D) − {x, y} such that x−→ u −→ v −→ y.
since the path of x −→ y passes through the cut vertices. Then S+ contains all cut vertices together with other dominating vertices.
Hence V − S+contains atleast two vertices one from A and other from B.
∴ γtc+(D)≤ n − 2.
Theorem 3.4. Let D be a directed wheel.Then γtc+(D)≤ n, n ≥ 3.
Proof. Let D be a directed wheel −→
Cn+ K1, the vertices of −→
Cndominates the vertex in
−→K1. Let S+ be a triple connected dominating set with atmost n-1 vertices, n ≥ 4. Then V − S+contains atleast one vertex from cycle and a vertex from −→K1.
Hence γtc+(D)≤ n.
Theorem 3.5. Let D be a directed comet.Then γtc+(D) = m, m ≥ 3
Proof. Let D be a directed comet of −−→Cm,n,where m and n are positive integers.An out-tree has obtained by central vertex of the directed star −−→K1,nwith the vertex of directed path −→Pm . Let S+be a triple connected dominating set having all the vertices in directed path and V − S+contains n vertices which dominate from the central vertex in directed path .
Hence γtc+(D) = m
4 Complementary connected triple connected domination number of digraph
4.1 Notation
1. γcctc+ - the set of vertices in complementary connected triple connected dominating set.
Definition 4.1. A subset S+ of V of a digraph D is said to be complementary connected triple connected dominating set if S+ is a dominating set and the induced subdigraph< S+ >is triple connected and < V − S+ >is weakly connected.The min- imum cardinality of complementary connected triple connected domination number and it is denoted by γcctc+ (D).
Observation 4.1. Let D be a directed path or directed cycle or orientation graph then γ+cctc(D)= γtc+(D)
Observation 4.2. If D is transitive tournament then γcctc+ (D) = γtc+(D)≤ n − 1 Theorem 4.1. Let D be a connected digraph with cut vertices.Then γcctc+ (D) ≤ n− 4+
Proof. Let D be a strongly connected digraph or unilateraly connected digraph with one or more components . Let S+be a triple connected dominating set and < V − S+ >
is weakly connected which has vertices belongs to same component.
Hence γcctc+ (D)≤ n − 4+
Theorem 4.2. Let D be a strongly connected bipartite digraph then γcctc+ (D) ≤ m + n− 1
Proof. Let D be a strongly connected bipartite digraph −−−→Km,n Let V1 and V2 are the partitions of V .Every vertices in V1 is adjacent from vertices in V2 or adjacent to vertices in V2to make the digraph is strong.
Let S+be triple connected dominating set with atleast three vertices from V1 and V2. Then < V − S+ >is weakly connected contains vertices from V1and V2 or it has exactly one vertex.
Hence γcctc+ (D)≤ m + n − 1 .
Theorem 4.3. Let D be a directed wheel.Then γcctc+ ≥ n − 4+ Proof. Let D be a directed wheel −→
Cn+K1 where −→
Cn is a directed cycle.since all the vertices of directed cycle dominate the vertex of −→K1which has maximum in-degree n.Let S+be a triple connected dominating set contains vertices which dominate the vertices of V − S+.Hence V − S+contains exactly one vertex from directed cycle and vertex in −→K1.
∴ < V − S+>is weakly connected.
Thus γcctc+ ≥ n − 4+
5 Conclusion
The concept of triple connected digraphs and domination in triple connected digraphs can be applied to physical problems such as flow networks with valves in the pipes and electrical networks, neural networks etc. They are applied in abstract representations of computer programs and are an invaluable tools in the study of sequential machines. In future this paper can be extended to studies of strong and weak domination in triple con- nected digraphs.
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