CN-edge domination in graphs Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2013, Том 15, Выпуск 2, С. 12-18

CN-EDGE DOMINATION IN GRAPHS

A. Alwardi, N. D. Soner

Let G = (V, E) be a graph. A subset D of V is called common neighbourhood dominating set (CN-domi-nating set) if for every v G V — D there exists a vertex u G D such that uv G E(G) and |r(u, v)| ^ 1, where |r(u, v)| is the number of common neighbourhood between the vertices u and v. The minimum cardinality of such CN-dominating set denoted by Yen (G) and is called common neighbourhood domination number (CN-edge domination) of G. In this paper we introduce the concept of common neighbourhood edge domination (CN-edge domination) and common neighbourhood edge domatic number (CN-edge domatic number) in a graph, exact values for some standard graphs, bounds and some interesting results are established.

Mathematics Subject Classification (2000): 05C69.

Key words: common neighbourhood edge dominating set, common neighbourhood edge domatic number, common neighbourhood edge domination number.

1. Introduction

By a graph G = (V, E) we mean a finite and undirected graph with no loops and multiple edges. As usual p = |V| and q = |E| denote the number of vertices and edges of a graph G, respectively. In general, we use (X} to denote the subgraph induced by the set of vertices X. N(v) and N[v] denote the open and closed neighbourhood of a vertex v, respectively. A set D of vertices in a graph G is a dominating set if every vertex in V — D is adjacent to some vertex in D. The domination number y(G) is the minimum cardinality of a dominating set of G. A line graph L(G) (also called an interchange graph or edge graph) of a simple graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge if and only if the corresponding edges of G have a vertex in common. For terminology and notations not specifically defined here we refer reader to [5]. For more details about domination number and its related parameters, we refer to [6], [9], and [10].

Let G be a simple graph G = (V, E) with vertex set V(G) = |vi, v2,..., vn}. For i = j, the common neighborhood of the the vertices vj and Vj, denoted by r(vj,vj), is the set of vertices, different from vj and Vj, which are adjacent to both Vj and Vj. A subset D of V is called common neighbourhood dominating set (CN-dominating set) if for every v £ V — D there exist a vertex u £ D such that uv £ E(G) and |r(u, v)| ^ 1, where |r(u, v)| is the number of common neighbourhood between the vertices u and v. The minimum cardinality of such CN-dominating set denoted by Ycn(G) and is called common neighbourhood domination number (CN-domination number) of G. The CN-domination number is defined for any graph. A common neighbourhood dominating set D is said to be minimal if no proper subset of D is common neighbourhood dominating set. If u £ V, then the CN-neighbourhood of u denoted by Ncn(u) is defined as Ncn(u) = {v £ N(u) : |r(u, v)| ^ 1}. The cardinality of Ncn(u) is denoted by degcn(u) in G, and Ncn[u] = Ncn(u) U {u}. The maximum and minimum common neighbourhood degree of a vertex in G are denoted respectively by Acn(G) and ¿cn(G). That

© 2013 Alwardi A., Soner N. D.

is Acn(G) = maxueV |Ncn(u)|, ¿cn(G) = minueV |Ncn(u)|. A subset S of V is called a common neighbourhood independent set (CN-independent set), if for every u G S,v i/ Ncn(u) for all v G S-{u}. It is clear that every independent set is CN-independent set. An CN-independent set S is called maximal if any vertex set properly containing S is not CN-independent set. The maximum cardinality of CN-independent set is denoted by and the lower CN-independence number icn is the minimum cardinality of the CN-maximal independent set. An edge e = uv G E(G) is said to be common neighbourhood edge (CN-edge) if |r(u, v)| ^ 1. For more details about CN-dominating set see [1]. The concept of edge domination was introduced by Mitchell and Hedetniemi [8]. Let G = (V, E) be a graph. A subset X of E is called an edge dominating set of G if every edge in E — X is adjacent to some edge in X.

In this paper analogue to the edge domination, we introduce the concept of common neighbourhood edge domination (CN-edge domination) in a graph and common neighbourhood edge domatic number (CN-edge domatic number) in a graph, exact values for the some standard graphs bounds and some interesting results are established.

2. CN-Edge Domination Number

Let G = (V, E) be a graph and f, e be any two edges in E. Then f and e are adjacent if they have one end vertex in common.

Definition 2.1. Two edges f and e are common neighbourhood adjacent (CN-adjacent) if f adjacent to e and there exist another edge g adjacent to both f and e.

Definition 2.2. A set S of edges is called common neighbourhood edge dominating .set (CN-edge dominating .set) if every edge f not in S is CN-adjacent to at least one edge f' G S. The minimum cardinality of such CN-edge dominating set is denoted by YCn(G) and called CN-edge domination number of G.

The CN-edge neighbourhood of f denoted by Ncn(f) is defined as Ncn(f) = {g G E(G) : f and g are CN-adjacent}. The cardinality of Ncn(f) is called the CN-degree of the edge f and denoted by degcn(f). The maximum and minimum CN-degree of edges in G are denoted respectively by Acn(G) and ¿cn(G). That is Acn(G) = maxje#(G) |Ncn(f)|, ¿^(G) = minf£E(G) |Ncn(f)|.

A common neighbourhood edge dominating set S is minimal if for any edge f G S, S — {f} is not CN-edge dominating set of G. A subset S of E is called CN-edge independent set, if for any f G S, f G Ncn(g), for all g G S — {f}. If an edge f G E be such that Ncn(f) = 0 then j is in any CN-dominating set. Such edges are called CN-isolated. The minimum CN-edge dominating set denoted by Ycn-set.

An edge dominating set X is called an CN-independent edge dominating set if no two edges in X are CN-adjacent. The CN-independent edge domination number Ycni(G) is the minimum cardinality taken over all CN-independent edge dominating sets of G. The CN-edge independence number ^«(G) is defined to be the number of edges in a maximum CN-independent set of edges of G. For a real number x; |_xj denotes the greatest integer less than or equal to x and |"x] denotes the smallest integer greater than or equal to x.

In Figure 1, E(G) = {a, b, c, d, e}. The minimal edge dominating sets are {b}, {a,c}, {a,d}, {a,e}, {b, c}, {b, d}, {b, e}. Therefore y'(G) = 1. The minimal CN-edge dominating sets are {a,b}, {a,c}, {a,d}, {a,e}. Therefore Ym(G) = 2. The edge a is CN-edge isolated but not edge isolated.

( a

A c 1 b ( * 1 e A

w d w

Fig. 1.

Observation 2.1. For any graph G with at least one edge, 1 ^ yfcn(G) ^ q.

If the graph G is triangle free and claw free and A(G) ^ 2, then for any two adjacent edge e and f there is no any edge adjacent both e and f, so we have the following proposition.

Proposition 2.1. Let G = (V,E) be a nontrivial graph with q edges. Then YCn(G) = q if and only G triangle free graph with A(G) ^ 2.

Hence it follows that

YCn(CP) = Ycn(Cp) = p and yCn(PP) = Ycn(pp-1) = p — 1

From the definition of line graph and the CN-edge domination the following Proposition is immediate.

Observation 2.2. For any graph G, we have Y'cn(G) = Yen(L(G)).

Proposition 2.2. For any complete graph Kp and Complete bipartite graph Kn have

p

Y'cn (KP) = [2 J and Y'cn(Kn,m)min{m,n}.

Proposition 2.3. For any wheel graph Wp of p vertices, we have

we

Y'cn(Wp) =

1 + P-3, if p = 0(mod 3); 1 + P-1, if p = 1(mod3);

1 + P-2

1 + 3 '

if p ^ 2(mod 3).

Obviously for any graph G any CN-edge dominating set is edge dominating set then the following proposition follows.

Proposition 2.4. For any graph G, Ycn(G) ^ y'(G).

Theorem 2.1. The CN-edge dominating set F is minimal if and only if for each edge f G F one of the following conditions holds

(i) Ncn(f ) n F =

(ii) there exist an edge g G E — F such that Ncn(g) n F = {f}.

< Suppose that F is a minimal CN-edge dominating set. Assume that (i) and (ii) do not hold. Then for some f G F there exist an edge g G Ncn(f ) n F and for every edge h G E — F, Ncn(h) n F = {f}. Therefore F — {f} is CN-edge dominating set contradiction to the minimality of F. Therefor (i) or (ii) holds.

Conversely, suppose for every f G F one of the conditions holds. Suppose F is not minimal. Then there exist f G F such that F — {f} is CN-edge dominating set. Therefore there exist an edge g G F — {f} such that g G Ncn(f). Hence f does not satisfy (i). Then f

must satisfy (ii). Then there exist an edge g G E — F such that Ncn(g) n F = {f}. Since F — {f} is CN-edge dominating set there exist an edge f' G F — {f} such that f' is CN-adjacent to g. Therefore f' G Ncn(g) n F and f' = f, a contradiction to Ncn(g) n F = {f}. Hence F is minimal CN-edge dominating set. >

Proposition 2.5. For any Graph G without any CN-isolated edges, if F is minimal CN-edge dominating set then E — F is CN-edge dominating set.

< Let F be minimal CN-edge dominating set of G. Suppose E — F is not CN-edge dominating set. Then there exist an edge f such that f G F is not CN-adjacent to any edge in E — F. Since G has no CN-isolated edges then f is CN-dominated by at least one edge in F — {f}. Thus F — {f} is CN-edge dominating set a contradiction to the minimality of F. Therefore E — F is is CN-edge dominating set. >

Proposition 2.6. For any graph G, any CN-independent edge set F is maximal CN-independent edge set if and only if it is CN-edge independent and CN-edge dominating set of G.

< Suppose F be maximal CN-independent set of G. Then for every edge f G E — F, the set F U {f} is not CN-independent, that is for every edge f G E — F, there is an edge g G F in such that f is CN-adjacent to g. Thus F is CN-edge dominating set. Hence F is both CN-edge independent and CN-edge dominating set of G.

Conversely, suppose F is both CN-edge independent and CN-edge dominating set of G. Suppose F is not maximal CN-edge independent set. Then there exist an edge f G E — F such that F U {f} is CN-independent, then there is no edge in F is CN-adjacent to f. Hence F is not CN-edge dominating set which is a contradiction. Hence F is maximal CN-independent set. >

Theorem 2.2. For any graph G, Ym(G) ^ q — A^G).

< Let f be an edge in G such that degcn(f) = A^G). Then E(G) — Ncn(f) is CN-edge dominating set. Hence Ym(G) ^ q — Acn(G). >

Theorem 2.3. For any Y'cn-set F of a graph G = (V, E), |E — F| ^ ^feF degcn(f) and the equality holds if and only if

(i) F is CN-independent edge

(ii) for every edge f G E — F, there exists only one edge g G F such that Ncn(f) nF = {g}.

< Since each edge in E — F is CN-adjacent to at least one edge of F. Therefore each edge in E — F contributes at least one to the sum of the CN-degrees of the edges of F. Hence

|E — F| < £ degcn(f).

f eF

Let |E — F| = ^f eF degcn(f) and suppose that F is not CN-independent edge. Clearly each edge in E — F is counted in the sum feF degcn(f). Hence if f and f2 are CN-adjacent edges, then f is counted in degcn(fi) and vice versa. Then the sum exceeds |E — F| by at least two, contrary to the hypothesis. Hence F must be CN-independent edge.

Now suppose (ii) is not true. Then Ncn(f) n F ^ 2 for f G E — F. Let f1 and f2 belong to Ncn(f) n F, hence J2feF degcn(f) exceed E — F by at least one since f counted twice, once in degcn(f1) and the other in degcn(f2). Hence if the equality holds then the condition (i) and (ii) must be true. The converse is obvious. >

Theorem 2.4. For any (p, q) graph G, A, (G)+1 ^ Ym(G) ^ q — ^ + qo, where q0 is the number of CN-isolated edges.

< From the previous proposition |E — F| ^ ^feF degcn(f) ^ Ym(G)Acn. Hence q — y£™(G) ^ Ycn(G)Acn. Therefore

< ycn(G).

Acn(G) + 1

(1)

Let G' = (E(G) — 1cn(G)) where /cn(G) is the set of CN-isolated edges of G. Let S be the maximal CN-independent set of edges of G'. Hence S is also CN-edge dominating set of G'. Since G' does not have CN-isolated edges E(G') — S is also CN-edge dominating set of G'. Therefore y^G') < |E(G') — S| = q(G') — $n(G'). But Ym(G') = Ycn(G) — qo and q(G') = q(G) — q0 and ^(G') = ^(G) — q0. Hence

Ycn(G) — qo < q(G) — qo — ($n(G) — qo).

Therefore

ycn(G) < q— &n + qo.

(2)

From (1) and (2) we have

Acn(G) +1

< Ycn(G) < q — #n + qo- >

q

q

3. CN-Edge Domatic Number

Definition 3.1. The maximum order of partition of the edges E(G) into CN-edge dominating sets is called CN-edge domatic number of G and denoted by dcn(G).

Observation 3.1. For any graph G, dcn(G) ^ d'(G), where d'(G) is the edge domatic number.

< Let G = (V, E) be a graph Since any partition of E(G) into CN-edge domination set is also partition of E(G) into edge dominating set. Hence dcn(G) ^ d'(G). >

Proposition 3.1. (i) For any cycle Cp and path Pp with p vertices dcn(Cp) = dcn(Pp) = 1.

(ii) For any complete bipartite graph Km,n, dcn(Km,n) = max{m, n}.

(iii) dcn(G) = 1 if and only if G has at least one CN-isolated edge.

Proposition 3.2. Let G be a complete graph Kp with p ^ 2 vertices. Then

tUG) = (P — 1, if n iseven;

p, if n is odd.

< If p is even, then Kp can be decomposed into p — 1 pairwise edge-disjoint linear factors. The edge set of each these factors is an edge dominating set in Kp. Since any two adjacent vertices in the complete graph are also CN-adjacent, then any edge dominating set is also CN-edge dominating set in Kp. Hence dcn(Kp) ^ p — 1. Suppose that dcn(Kp) ^ p and consider Z is an CN-edge domatic partition of Kp with p classes the mean value of the orders of these classes has at most p-1. This implies that at least one of the classes has at most [p-1] = § — 1 edges. But the partition covers at most p — 2 vertices, there are two vertices no edge in Z incident to any of the two vertices. Hence there is no edge in Z CN-adjacent to the edge joining these vertices. Therefore dcn(Kp) = p — 1 if p is even.

Now let p is odd. By labeling the vertices of Kp by X1, X 2, . . . , Xp. We can make the following domatic partition of the vertices E1 ,...,Ep, where Ei = xi+jxi-j+1, where j = 1,..., p-1 and the scripts will be taken modulo p. Hence d'cn (Kp) ^ p. If we suppose that d'cn(Kp) ^ p + 1, then in the same way of the first case we can prove that there exist an CN-edge domatic partition one of whose classes has at most p-3 edges this set cover at most p — 3 vertices and it is not an CN-edge dominating set a contradiction. Hence dCn(Kp) = p if p is odd. >

Theorem 3.1. For any graph G with q edges, d'cn(G) ^ s, qG).

< Assume that d'cn(G) = d and {D1 ,D2,..., Dd} is a partition of E(G) into d CN-edge dominating sets, clearly |Di| ^ j'cn(G) for i = 1,2,...,d and we have q = ^= |Di| ^ dS'cn(G).

Hence d'cn(G) < scnjoj. >

References

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Received April 10, 2012.

Alwardi Saleh Anwar

Department of Studies in Mathematics,

University of Mysore

Mysore 570006, India

E-mail: [email protected]

Soner Nandappa d.

Department of Studies in Mathematics,

University of Mysore

Mysore 570006, India

E-mail: [email protected]

CN-РЕБЕРНОЕ ДОМИНИРОВАНИЕ В ГРАФАХ Алварди А., Сонер Н.

Пусть G = (V, E) — граф. Подмножество D множества V называется реберно доминирующим множеством с общей окрестностью (CN-реберно доминирующим множеством), если для любой вершины v G V — D существует вершина u G D такая, что uv G E(G) и |г(и, v)| ^ 1, где |г(и, v)| — множество общих соседей вершин u и v. Наименьшая мощность такого CN-реберно доминирующего множества обозначается ycn(G) и называется реберно доминирующим числом с общей окрестностью (CN-реберно доминирующим числом) графа G. В данной статье вводятся понятия реберно доминирующего числа с общей окрестностью и реберно доматического числа с общей окрестностью (CN-реберно доматического числа) в графе, найдены их точные значения в некоторых стандартных графах, установлены границы и некоторые интересные результаты.

Ключевые слова: реберно доминирующее множество с общей окрестностью, реберно доматическое число с общей окрестностью, реберно доминирующее число с общей окрестностью.