Secure Doubly Connected Domination in Graphs

Int. Journal of Math. Analysis, Vol. 8, 2014, no. 32, 1571 - 1580HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijma.2014.46175

Secure Doubly Connected Domination in GraphsBenjier H. Arriola1

Mathematics DepartmentBasilan State College

Isabela City, 7300 Basilan Province, Philippines

Sergio R. Canoy, Jr.

Department of Mathematics and StatisticsMindanao State University-Iligan Institute of Technology

Tibanga Highway, Iligan City, Philippines

Copyright c© 2014 Benjier H. Arriola and Sergio R. Canoy, Jr. This is an open accessarticle distributed under the Creative Commons Attribution License, which permits unre-stricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Abstract

In this paper, we introduced a new domination parameter calledsecure doubly connected domination number of a graph. Specifically,if G is a connected graph, we denote by γscc(G) the secure doublyconnected domination number of G. We obtained several properties ofthe secure doubly connected dominating sets of a graph. In particular,we characterize all graphs G such that γscc(G) = n, where n is the orderof G. Some bounds and exact values of the secure doubly connecteddomination numbers of some graphs are also determined.

Mathematics Subject Classification: 05C69

Keywords: domination, doubly connected domination, secure domination,secure doubly connected domination

1This research is partially funded by the CHED-FDP II, Philippines

1572 B. H. Arriola and S. R. Canoy, Jr.

1 Introduction

The concept of doubly connected domination was introduced in 2006 by Cyman,Lemanska and Raczek [8]. This parameter has been characterized for somebinary operations [1]. Another domination parameter is called the securedomination which has been introduced and explored in the papers of Cockayneet al. in [2, 3, 4, 5, 6, 7] and Mynhardt et al. in [7, 11, 12]. Go and Canoy[9] characterized total secure dominating set in the join of two graphs. Acombination of some of these concepts gives rise to a new variant of dominationcalled secure doubly connected domination.

Let G be a graph. A set S ⊆ V (G) is a dominating set of G if for everyv ∈ V (G)\S, there exists u ∈ S such that uv ∈ E(G), i.e., NG[S] = V (G).The domination number of G, denoted by γ(G), is the smallest cardinality ofa dominating set of G. A dominating set S of G with |S| = γ(G) is called aγ-set.

Let G be a connected graph. A dominating set S ⊆ V (G) is called aconnected dominating set of G if the subgraph 〈S〉 induced by S is connected.The connected domination number of G, denoted by γc(G), is the smallestcardinality of a connected dominating set of G. A connected dominating set Sof G with |S| = γc(G) is called a γc-set.

Let G be a connected graph. A connected dominating set S ⊂ V (G) is calleda doubly connected dominating set of G if the subgraph 〈V (G)\S〉 induced byV (G)\S is connected. The doubly connected domination number of G, denotedby γcc(G), is the smallest cardinality of a doubly connected dominating set ofG. A doubly connected dominating set S of G with |S| = γcc(G) is called aγcc-set.

Let G be a graph. The set S ⊆ V (G) is a secure dominating set of Gif for each u ∈ V (G)\S, there exists a vertex v ∈ S such that uv ∈ E(G)and (S\{v} ∪ {u}) is a dominating set. The secure domination number of G,denoted by γs(G), is the smallest cardinality of a secure dominating set of G.A secure dominating set S of G with |S| = γs(G) is called a γs-set.

Let G be a connected graph. A subset S of V (G) is a secure doubly connecteddominating set of G if S = V (G) or S is a doubly connected dominating setof G such that for each u ∈ V (G)\S, there exists v ∈ S with uv ∈ E(G) and(S\{v})∪{u} is a doubly connected dominating set. The secure doubly connecteddomination number of G, denoted by γscc(G), is the smallest cardinality ofa secure doubly connected dominating set of G. A secure doubly connecteddominating set S of G with |S| = γscc(G) is called a γscc-set.

For a graph G, we denote ∆(G) as the maximum degree of G and m as thesize of G. For any undefined terms, see [10].

Secure doubly connected domination... 1573

2 Preliminary ResultsNote that every secure doubly connected dominating (proper) subset in a graphis both a doubly connected dominating set and a secure connected dominatingset. We state this observation formally.

Remark 2.1 For any connected graph G of order n,

1 ≤ γc(G) ≤ γcc(G) ≤ γscc(G) ≤ n and γs(G) ≤ γsc(G) ≤ γscc(G).

Lemma 2.2 Let G be a connected graph of order n ≥ 3 and let S be a securedoubly connected dominating set of G. Then the following hold:

(i.) Every cut-vertex of G is in S.

(ii.) Every leaf of G is in S.

(iii.) If G contains two adjacent non-cut-vertices, then γscc(G) ≤ n− 1.

(iv.) If G contains two non-adjacent vertices (i.e., G is not complete), then|S| ≥ 2.

Proof : (i.) Let v be a cut-vertex of G and let S be a secure doubly connecteddominating set of G. Suppose v ∈ V (G)\S. Since 〈S〉 is connected, S iscontained in some component of 〈V (G)\{v}〉. Thus, V (G)\S contains allvertices of other components of 〈V (G)\{v}〉. Since v /∈ S, S cannot be adominating set of V (G), contrary to our assumption of S.

(ii.) Let v be a leaf of G and let S be a secure doubly connected dominatingset of G. Suppose v /∈ S. Since S is a dominating set, then there exists exactlyone vertex x ∈ S such that vx ∈ E(G). Since 〈(S\{x}) ∪ {v}〉 is disconnected,(S\{x}) ∪ {v} is not a doubly connected dominating set of G, a contradiction.Thus, v ∈ S.

(iii.) Let u, v ∈ V (G) such that uv ∈ E(G) and both u and v are non-cut-vertices. Let S1 = V (G)\{u} and S2 = V (G)\{v}. Then S1 and S2 are doublyconnected dominating sets of G. Since (S1\{v}) ∪ {u} = S2, S1 is a securedoubly connected dominating set of G. Hence, γscc(G) ≤ |S1| = n− 1.

(iv.) Let u and v be two non-adjacent vertices of G. Suppose |S| = 1.Let S = {x} be a secure doubly connected dominating set of G. Since Sis a dominating set, u 6= x, v 6= x and ux, vx ∈ E(G). Sinceuv /∈ E(G), (S\{x})∪{u} = {u} is not a dominating set, a contradiction. Thus,|S| ≥ 2. �


3 ResultsThe next results characterize all graphs G such that γscc(G) is 1, 2, or n, wheren is the order of G.

Theorem 3.1 Let G be a connected graph of order n ≥ 1. Then γscc(G) = 1if and only if G is a complete graph.

Proof : Suppose γscc(G) = 1. Suppose further that G is not a completegraph. Then there exists two non-adjacent vertices in G. By Lemma 2.2 (iv),γscc(G) ≥ 2, contrary to our assumption.

For the converse, let S = {u}, where u ∈ V (G). Clearly S is a doublyconnected dominating set of G. Let v ∈ V (G)\{u}. Then uv ∈ E(G) and(S\{u}) ∪ {v} is a doubly connected dominating set of G. This implies that Sis a secure doubly connected dominating set of G. Thus, γscc(G) = 1. �

Theorem 3.2 Let G be a connected non-complete graph. Then γscc(G) = 2 ifand only if there exists a connected graph H such that G = K2 +H.

Proof : Suppose γscc(G) = 2. Let S = {x, y} be a secure doubly connecteddominating set of G. Take H = 〈V (G)\S〉. Since S is a doubly connecteddominating set of G, 〈S〉 and H are both connected (hence 〈S〉 = K2). Next,let u ∈ V (H). Since S is a dominating set of G, ux ∈ E(G) or uy ∈ E(G).Suppose ux ∈ E(G). Since (S\{x}) ∪ {u} = {u, y} is a doubly connecteddominating set of G, 〈{u, y}〉 is connected. This implies that uy ∈ E(G). Thismeans that every vertex in H is adjacent to both x and y. Thus, G = K2 +H.

Conversely, suppose G = K2 + H, where H is connected. ThenS = V (K2) = {x, y} is a doubly connected dominating set of G. Let u ∈ V (H).Then (S\{x}) ∪ {u} is also a doubly connected dominating set of G. Thus,γscc(G) ≤ 2. Since G is not a complete graph, γscc(G) ≥ 2 by Theorem 3.1.Hence, γscc(G) = 2. �

The next result characterized the upper bound of the secure doubly con-nected domination number.

Theorem 3.3 Let G be a connected graph of order n ≥ 3. Then γscc(G) = nif and only if for every two adjacent vertices in G, at least one of these is acut-vertex.

Proof : Suppose γscc(G) = n. Then S = V (G) is a γscc-set of G. Let u, v ∈ V (G)such that uv ∈ V (G). Suppose both u and v are not cut-vertices. Then byLemma 2.2 (iii), |S| ≤ n− 1 which is a contradiction.

Conversely, suppose that for any two adjacent vertices x, y in G, at least oneof these is a cut-vertex. If x is a cut-vertex, then 〈V (G)\{x}〉 is not connected.If x is not a cut-vertex, then 〈V (G)\{y}〉 is not connected. Thus, V (G) is aγscc-set of G. �


Corollary 3.4 Let Tn, Pn, and K1,n−1 be the tree, path, and star, respectively,on n ≥ 3 vertices. Then

γscc(Tn) = γscc(Pn) = γscc(K1,n−1) = n.

Proof : For every two adjacent vertices of Tn, Pn, or K1,n−1, at least one is acut-vertex. By Theorem 3.3, γscc(Tn) = γscc(Pn) = γscc(K1,n−1) = n. �

The next lemma is a proposition in [8].

Lemma 3.5 [8] Let S be a γcc-set of a connected graph G of order n ≥ 3.Then every cut-vertex and support of G is in S.

Theorem 3.6 The difference γscc − γcc can be made arbitrarily large.

Proof : Consider a graph G obtained from a wheel Wn+1, where n ≥ 3 byadding a vertex x and an edge xy where y is the central vertex of Wn+1 (seeFigure 1). Since y is a cut-vertex and x is a leaf, by Lemma 3.5, x and y

................................................................................................... ....................................

.........

........

........

........

........

........

........

........

........

........

........

....

....................................

...................................................................................................

....................................

................................................................................................................................. ...................................................................................................

....................................

.........

........

........

........

........

........

........

........

........

........

........

........................................

...................................................................................................

....................................

....................................

.................................... ....................................

.................................................................................................................................

...........................................

..........................................

..........................................

..

.................................................................................................................................

..................................................................................................................................................................................................................................................................

...........................................

..........................................

..........................................

..

.................................................................................................................................

...........................................................................................................................................................................................

..........................................................................................................................................................................

...un+1

u1

u2

u3 u4

u5

u6

u7

yx

Figure 1: A graph G with γcc(G) < γscc(G)

are both in a γcc-set of G. Since {x, y} is a connected dominating set and〈V (G)\{x, y}〉 is connected, {x, y} is a doubly connected dominating set of G.Since there exists no doubly connected dominating set of G with cardinality1, γcc(G) = 2. Next, since G has two adjacent non-cut-vertices, by Lemma2.2 (iii), γscc(G) ≤ |V (G)| − 1 = n + 2. Now, let S1 be a secure doublyconnected dominating set of G such that |S1| < n+ 2. Thus, |V (G)\S1| ≥ 2.By Lemma 2.2 (i) and (ii), x, y ∈ S1. Let C = [u1, u2, ..., un+1, u1] and letui ∈ V (G)\S1, where i ∈ {1, 2, ...n+ 1}. Since C is a cycle, it can be assumethat u1 ∈ V (G)\S1. Consider the following cases:

Case 1. |V (G)\S1| = 2Then either un+1 ∈ V (G)\S1 or u2 ∈ V (G)\S1. Suppose u2 ∈ V (G)\S1.

Then 〈(S\{un+1}) ∪ {u1}〉 and 〈(S\{y}) ∪ {u1}〉 are disconnected, contrary


to our assumption that S1 is a secure doubly connected dominating set ofG. Similarly, if un+1 ∈ V (G)\S1, 〈(S\{u2}) ∪ {u1}〉 and 〈(S\{y}) ∪ {u1}〉 aredisconnected which also contradicts our assumption.

Case 2. |V (G)\S1| > 2If un+1 ∈ S1, then u2, u3 ∈ V (G)\S1. This implies that 〈(S\{y}) ∪ {u2}〉 is

disconnected, contrary to our assumption. If u2 ∈ S1, then un+1, un ∈ V (G)\S1.This implies that 〈(S\{y}) ∪ {un+1}〉 is disconnected which is also a contradic-tion. If u2, un+1 /∈ S1, then 〈(S\{y}) ∪ {u1}〉 is disconnected, also contradictsour assumption.

Hence, γscc(G) ≥ n+ 2. Therefore, γscc(G)− g(G) = (n+ 2)− 2 = n. �

In [8], Cyman et al. characterized the doubly connected domination numberof a connected graph in terms of its maximum degree, order, and size. Thischaracterization is given in the next theorem.

Theorem 3.7 [8] Let G be any connected graph of order n ≥ 2 and size m.Then

n

∆(G) + 1 ≤ γcc(G) ≤ 2m− n+ 1

with equality for the lower bound hold if and only if γcc(G) = 1 and equality forthe upper bound hold if and only if G is a tree.

A similar result for the secure doubly connected domination number can beobtained.

Theorem 3.8 Let G be any connected graph of order n ≥ 2 and size m. Thenn

∆(G) + 1 ≤ γscc(G) ≤ 2m− n+ 2

with equality for the lower bound holds if and only if γscc(G) = 1 and equalityfor the upper bound holds if and only if G is a tree.

Proof : By Remark 2.1 and Theorem 3.7, n∆(G)+1 ≤ γcc(G) ≤ γscc(G) and

γscc(G) ≤ n = 2(n − 1) − n + 2. Since G is connected, m ≥ n − 1. Thus,γscc(G) ≤ 2(n− 1)− n+ 2 ≤ 2m− n+ 2.

Next, suppose that γscc(G) = 1. Then by Theorem 3.1, G is a completegraph. This implies that ∆(G) = n− 1. Thus, n

∆(G)+1 = n(n−1)+1 = 1 = γscc(G).

Conversely, suppose n∆(G)+1 = γscc(G). By Theorem 3.7 and Remark 2.1,

γscc(G) = g(G) = n∆(G)+1 .

If G is a tree, then m = n − 1 and, by Theorem 3.3, γscc(G) = n =2(n− 1)− n+ 2 = 2m− n+ 2. Conversely, let γscc(G) = 2m− n+ 2. Again,


by Remark 2.1, 2m − n + 2 ≤ n. This implies that m ≤ n − 1. Since G isconnected, m ≥ n− 1. Thus, m = n− 1 and hence, G is a tree. �

The corona of two graphs G and H, denoted by G◦H, is the graph obtainedby taking one copy of G of order n and n copies of H, and then joining thei-th vertex of G to every vertex in the i-th copy of H. For every v ∈ V (G),we denote by Hv the copy of H whose vertices are joined or attached to thevertex v. In [1], the doubly connected domination for the corona of two graphshas been characterized and it is given in the following theorem. We need thistheorem to prove the next result.

Theorem 3.9 [1] Let G be a connected graph and H be any graph. Then anonempty set C ⊂ V (G ◦H) is a doubly connected dominating set of G ◦H ifand only if there exists a vertex v of G such that

C = V (G)⋃ ⋃

u∈V (G)\{v}V (Hu)

⋃ (V (Hv)\T v) ,

where 〈T v〉 is a connected subgraph of Hv.

The next theorem is a characterization of a secure doubly connected domi-nation set in the corona of two graphs.

Theorem 3.10 Let G be connected graph and H be any graph. Then anonempty set C ⊆ V (G ◦ H) is a secure doubly connected dominating setof G ◦H if and only if

C = V (G)⋃ ⋃

u∈V (G)\{v}V (Hu)

⋃ (V (Hv)\T v) ,

where either T v = ∅ or 〈T v〉 is a connected proper subgraph of Hv andfor each y ∈ T v, there exists w ∈ V (Hv)\T v such that yw ∈ E(Hv) and〈(T v\{y}) ∪ {w}〉 is connected.

Proof : Let C ⊆ V (G ◦ H) be a secure doubly connected dominating set ofG ◦ H. If C = V (G ◦ H), then T v = ∅. Suppose S ⊂ V (G ◦ H). Then byTheorem 3.9,

C = V (G)⋃ ⋃

u∈V (G)\{v}V (Hu)

⋃ (V (Hv)\T v)

such that 〈T v〉 is a connected subgraph of Hv. Suppose T v = Hv. ThenC ∩ NHv = {v}. Since S is a secure doubly connected dominating set ofG ◦H, (C\{v}) ∪ {x} for each x ∈ T v is a doubly connected dominating set


of G ◦H. This is not possible since 〈(C\{v}) ∪ {x}〉 is not connected. Thus,T v ⊂ Hv. Let y ∈ V (G ◦ H)\C. Then y ∈ T v. Since C is a secure doublyconnected dominating set of G◦H, there exists w ∈ C such that wy ∈ E(G◦H)and (C\{w}) ∪ {y} is a doubly connected dominating set of G ◦H. Since vis a cut-vertex in G ◦ H, w 6= v. This implies that w ∈ V (Hv)\T v. Thus,〈(T v\{y}) ∪ {w}〉 is connected.

The converse is clear. �

Corollary 3.11 Let G be a connected non-trivial graph of order n ≥ 2 and letH be any graph of order m. Then γscc(G ◦H) ≤ m+mn− q, where

q = max{|V (C)| − γsc(C) : C is a component of H}.

Proof : Let v ∈ V (G) and let C be a component of Hv with q = |V (C)|−γscc(C)and let DC be a γscc-set of C. Set T v = V (C) \DC . Then D = V (Hv) \ T v =(V (Hv) \ V (C)) ∪ DC and |D| = m − q. If DC = V (C), then D = V (Hv)and S = V (G ◦ H) is a secure doubly connected dominating set of G ◦ H.So suppose DC 6= V (C). Since DC is a doubly connected dominating set ofC, 〈T v〉 is a connected proper subgraph of C (and hence of Hv). Moreover,for each y ∈ T v, there exists w ∈ DC ⊆ D such that (DC \ {w}) ∪ {y} is adoubly connected dominating set of C. This implies that 〈(T v \ {y}) ∪ {w}〉 isa connected subgraph of C (and hence of Hv). Therefore, by Theorem 3.10,

S = V (G)⋃( ⋃u∈V (G)\{v}

V (Hu))⋃

D is a secure doubly connected dominating

set of G ◦H. Accordingly,

γscc(G◦H) ≤ |S| = m+ (n−1)m+ |D| = m+mn−m+m− q = m+mn− q.

This proves the assertion. �

Remark 3.12 The strict inequality in Corollary 3.11 can be attained.However, the given upper bound is sharp.

To see this, consider the graphs G1 = K2 ◦H1 and G2 = K2 ◦H2 as shown inFigure 2. The shaded vertices in G1 form a γscc-set of G1 and the shaded verticesin G2 form a γscc-set of G2. Consider the components CHu

1= 〈{u1, u2, u3}〉 and

CHv2

= 〈{v1, v2, v3, v4, v5, v6}〉 of Hu1 and Hv

2 , respectively. Then E1 = {u1} is asecure connected dominating set of CHu

1and E2 = {v1, v2, v3, v5, v6} is a secure

connected dominating set of CHv2. Moreover, γscc(G1) = 8 = 2 + 2(4)− (3−

1) = |V (K1) + |V (K1)||V (H1)| − (|V (H1)| − |E1|) and γscc(G2) = 13 < 15 =2 + 2(7)− (6− 5) = |V (K2)|+ |V (K2)||V (H2)|+ (|V (H2)| − |E2|).


......................................................................... ....................................

.........................................................................

....................................

....................................

....................................

....................................

.........................................................................

.........................................................................

.........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

..

................................................................................................................................................................................

...............................................................................................................................................

...................................................

..................................................

.......

................................................................................................................................................................................

.........................................................................

....................................

......................................................................... ....................................

....................................

....................................

....................................

.........................................................................

.........................................................................

.........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

..

................................................................................................................................................................................

.................................................................

................................................................

..............

............................................................................................................

................................................................................................................................................................................ ...........................

u1

u2

u3

u

Hu1

G1 : •

•

•

•

•

•

•

• ................................................................

....................................

........................................... ....................................

.....................................

....................................

....................................

........................................... ....................................

.........

........

........

........

........

........

........

.......

....................................

....................................

....................................

....................................

.........................................................................

.........................................................................

.........................................................................................................................................................................................................

................................................................................................................................................................................

...............................................................................................................................................

.......................................................................................................................................................................................

................................................................................

...................

.....................................

....................................

....................................

....................................

...........

.........................................................................................................................................................................................................

.........

........

........

........

........

........

........

.......

....................................

...........................................

....................................

......................................................................... ....................................

...........................................

....................................

.........

........

........

........

........

........

........

.......

....................................

....................................

....................................

....................................

.........................................................................

.........................................................................

.........................................................................................................................................................................................................

................................................................................................................................................................................

.................................................................

................................................................

....................................................................................................................

....................................................................................................................................................................................

............................................................................................................................................................

......................................................................................................................................................................................................... ...........................

v6

v5v4

v3

v2

v1

v

Hv2

G2 : •

•

•

•

••

••

••

•

•

•

Figure 2: Graphs with γscc(G ◦H) ≤ |V (G)|+ |V (G)||V (H)| − (|V (C)|+ γcc(C)), where Cis component of H

References[1] B. H. Arriola and S. R. Canoy, Jr., Doubly Connected Domination in

the Corona and Lexicographic Product of Graphs, Applied MathematicalSciences. 8(2014), 1521 − 1533

[2] S. Benecke, E. Cockayne and C. Mynhardt, Secure Total Domination inGraphs. Utilitas Math. 74(2007), 247-259.

[3] A. P. Burger, E. J. Cockayne, W. R. Grundlingh, C. M. Mynhardt, J. H.van Vuuren, and W. Winterbach, Finite order domination in graphs. J.Combin. Math. Combin. Comput. 49 (2004), 159-175.

[4] A. P. Burger, E. J. Cockayne, W. R. Grundlingh, C. M. Mynhardt, J. H.van Vuuren, and W. Winterbach, Infinite order domination in graphs. J.Combin. Math. Combin. Comput. 50 (2004), 179-194.

[5] E. J. Cockayne, Irredundance, secure domination and maximum degree intrees. Discrete Math. 307 (2007), 12-17.

[6] E. J. Cockayne, O. Favaron, and C. M. Mynhardt, Secure domination,weak Roman domination and forbidden subgraphs. Bull. Inst. Combin. Appl.39 (2003), 87-100.

[7] E. J. Cockayne, P. J. P. Grobler, W. R. Grundlingh, J. Munganga, and J.H. van Vuuren, Protection of a graph. Util. Math. 67 (2005), 19-32.

[8] J. Cyman, M. Lemanska, J. Raczek, On the Doubly Connected DominationNumber of a Graph, Cent. Eur. J. Math. 4 (2006), 34-45.


[9] C. E. Go and S. R. Canoy, Domination in the Corona and Join of Graphs.International Mathematical Forum, Vol. 6, 2011, no.16, 763-771.

[10] F. Harary, Graph Theory, Addison-Wesley Publishing Company, Inc.,Philippines copyright 1969.

[11] W. Klostermeyer and C. Mynhardt, Secure Domination and Secure TotalDomination in Graphs. Discuss. Math. Graph Theory 28(2008), 267-284.

[12] C. M. Mynhardt, H. C. Swart, and E. Ungerer, Excellent trees and securedomination. Util. Math. 67 (2005), 255-267.

Received: June 15, 2014

Secure Doubly Connected Domination in Graphs

Documents

Transcript of Secure Doubly Connected Domination in Graphs