Independent Domination in Line Graphs

International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 1 ISSN 2229-5518

M. H. Muddebihal and D. Basavarajappa

Abstract - For any graph G , the line graph LGH is the intersection graph. Thus the vertices of LG are the edges of G , with two vertices of adjacent whenever the corresponding edges of G are. A dominating set D is called independent dominating set of LG , if D is also independent. The independent domination number of LG denoted by i L G , equals min{DD ; is an independent dominating set of LG} . A domatic partition of LG is a partition of VLG , all of whose classes are dominating sets in LG . The maximum number of classes of a domatic partition of LG is called the domatic number of LG and denoted by d L G . In this paper many bounds on were obtained in terms of elements of G , but not in terms of elements of . Further we develop its relationship with other different domination parameters. Also we introduce the concept of domatic number in LG .

Subject classification Number: AMS05-C69, C70.

Key words: Domatic number, Domination number, Graph, Independent dominating set, Independent domination number, Line graph and Odd graph. ——————————  ——————————

1 INTRODUCTION: In this paper, we follow the As usual ()GGis the minimum notations of [1]. All the graphs considered here are (maximum) degree. The degree of an edge e uv simple, finite, non-trivial, undirected, and connected. of G is defined by dege deg u deg v 2 and As usual pVand qE denote the number of '' GG is the minimum (maximum) degree vertices and edge of a graph G respectively. In among the edges of G . general we use X to denote the sub graph induced A spider is a tree with the property that the by the set of vertices X and Nv and Nvdenote removal of all end paths of length two of T results in the open and closed neighborhoods of a vertex v . The an isolated vertex, called the head of the spider Let k 2 be an integer. The odd graph Ok is notation 01GG is the minimum number of the graph whose vertex set is V and in which two vertices (edges) in a maximal independent set of vertex vertices are adjacent if and only if they are disjoint as (edge) of G . sets. Let deg v is the degree of vertex v and a A line graph LG is the graph whose vertex of degree one is called an end vertex and its neighbor is called a support vertex. vertices correspond to the edges of G and two vertices of LG are adjacent if and only if the corresponding ———————————————— edges in G are adjacent. M.H.Muddebihal, Department of Mathematics, Gulbarga University, A set D V is said to be dominating set of Gulbarga – 585106, Karnataka, India. E-mail: [email protected] G , if every vertex not in D is adjacent to a vertex in

D. Basavarajappa, Department of Mathematics, Gulbarga University, D . The domination number of G , denoted by G , Gulbarga – 585106, Karnataka, India. and Department of Mathematics, S.T.J. Institute of Technology, is the minimum cardinality of a dominating set. Ranebennur - 581115, Karnataka, India. A set D is an independent dominating set of E-mail: [email protected] G , if D is also independent. The independent IJSER © 2012 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 2 ISSN 2229-5518 domination number of G , denoted by iGis the The following Theorem relates domination and minimum cardinality of an independent dominating independent domination in . set. The concept of domination is now well studied in Theorem 2: For any connected pq, – graph, graph theory (see [2], [3]). ' (L ( G )) i ( L ( G )) q . Equality holds ifG C . Analogously, a set DVLG( ( )) is said to 4 Proof: Suppose D{ v , v , v ,..., v } V ( L ( G )) be be dominating set of LG , if every vertex not in D 1 2 3 n the set of vertices which covers all the vertices is adjacent to a vertex in D . The domination number in . Then D is a minimal - set of . of , denoted by LG is the minimum Further, if the sub graph D contains the set of cardinality of a dominating set in . ' vertices vi ,1 i n , such that degvi 0 . Then D A dominating set D of is said to be ' itself is an independent dominating set of . independent dominating set if D is also independent. ' ' The independent domination number of , Otherwise, SDI , where DD and I V(L(G)) D forms a minimal independent denoted by i L G is the minimum cardinality of an dominating set of .Since VLGEG , it independent dominating set in . follows that D S q . Therefore, A domatic partition in is a partition of . VLG , all of whose classes are dominating sets in For equality, suppose G C4 , then in this . The maximum number of classes of a domatic case D S2 q / 2 . Clearly, it follows that, partition of is called the domatic number of (L ( G )) i ( L ( G )) q . and is denoted by d L G . The concept of In the following Theorems we give the upper domatic number in G was introduced by Cockeyne et.al, [4]. bounds for independent domination number of LG . In this paper, many bounds on were Theorem 3: If every support vertex of tree is adjacent to at qm obtained. Also their relationships with other least one end edge, then i( L ( T )) 1, where domination parameters were obtained. Further, we 2 introduce the concept of domatic number of , m is the number of end edges inT . Equality holds for exact values of domatic number were obtained for star K1,p 1 . some standard graphs. Also bound for domatic Proof: Let F{ e1 , e 2 , e 3 ,..., en } be the set of all end number is also obtained. edges in T such that Fm. Now without loss of 2 RESULTS: ' generality, sinceVLTET , let SFH , Initially we list out independent domination ' number of for some standard graphs. where FF and HVLTF( ( )) , such that HNF[] Theorem 1: be the minimal set of vertices which covers all the vertices in LT . Clearly set of the vertices of a a. i( L ( Cp )) p / 3 . sub graph S is independent, then by the above b. i( L ( K1,n )) 1. argument S is a minimal independent dominating set c. i( L ( Kmn, )) n for mn. q m d. i( L ( K )) p / 2 . of . Clearly it follows that, S 1 . p 2 (p 2) e. i( L ( W )) . p 3 Therefore, .

Suppose T K1, p 1 . Then in this case, qm. Proof: Assume that degeq 1. Now in ,

Since LTKp and S 1, it follows that VLGEG , suppose F{ v12 , v ,..., vn } be qm the set of all end vertices in . Then there exists a i( L ( T )) 1. 2 vertex set D{ v1 , v 2 , v 3 ,..., vk } V ( L ( G )) F ,

which covers all the vertices in . Further, if the Theorem 4: For any connected graph G , i( L ( G ))1 ( G ) . sub graph D is independent, then D forms a

Proof: Suppose J{ e1 , e 2 , e 3 ,..., en } be the maximum minimal independent dominating set of set of edges in G such that N()() eij N e with D 2 , a contradiction. for1 i j n . Then j forms maximal independent Suppose . Then in , since set of edges with jG(). 1 , there exists a vertex SinceVLGEG , there exists an independent v V (L(G)) which covers all vertices in and ' ' set DJH , where JJ and v D . Clearly, D itself is a minimal independent HVLGJ( ( )) such that HNJ[], which dominating set of . Therefore, D 1and covers all the vertices in LG . Clearly, D forms a hence . minimal independent dominating set of LG and it Theorem 7: For any tree T, i ( L ( T )) ( p 1) / 2. follows that D J . Therefore, . Equality holds if and only if T is isomorphic to a spider. ' Proof: Let F{ v , v , v ,..., v } V ( L ( G )) be the Theorem 5: For any connected pq, – graph G , 1 2 3 i ' ' set of all vertices with FNF(), where F is the i( L ( G )) q ( G ) . set of end vertices of LT . Suppose Proof: Suppose C{ v1 , v 2 , v 3 ,..., vk } be the set of all ' non end vertices in G . Then there exists at least one HVLTF( ( )) and HNF(). Then ' vertex v C which is incident with at least one edge DIF where I H , covers all the vertices in ' ' eG() in G . Now without loss of generality in . Further, if the sub graph IF is LG , suppose H{ v , v , v ,..., v } be the set of 1 2 3 n independent, then forms a minimal all end vertices in and if VLGHI . independent dominating set of . Clearly, it Then there exists a subset D I in such that ' follows that I F( p 1) / 2 . Therefore, the sub graph D is independent. Clearly, D is an i i( L ( T )) ( p 1) / 2. Suppose T is isomorphic to ' - set of . It follows that, D q() G and q spider. Then in this case DFor D . hence . 2 Since, for any tree T , qp1 and each p in T is Corollary 1: For any tree T containing an odd, it follows that, . ' ' edge eT(), i(()() L T q T if and only if diam T 3.

Theorem 6: For any connected pq, – graph G , In the following Theorem, we list out the domatic number for some standard graphs. i( L ( G )) 1. If and only if G contains an edge e E(G) such that degeq 1. Theorem 8:

I. For any complete graph K p ,

VLG which is evidently a dominating set d L Kp p 1, if p is even.

= p , if p is odd. in LG . Now let k0 2 and suppose that the result

is true for kk0 1 . Consider a line graph in II. For any complete bipartite graph K with pp12, which the degree of each vertex is at least k0 . Let p12 p p vertices, VLG0 be a maximal (with respect to the set d L K pp, = max pp12, . 12 inclusion) independent set of vertices of . This III. For any cycle C , p set is dominating; otherwise a vertex could be added to it without violating the independence, which would be d L Cp 3, if p is divisible by 3. a contradiction with the maximality of . Let = 2, otherwise.

IV. For any path Pp , LG0 be a line graph obtained from by deleting all vertices of . Each vertex of d L Pp 2 .

is incident at most with one vertex of V0 , V. For any star K1,n with 1 np vertices, therefore each vertex of has the degree at least d L K1,n n . k 1. According to the hypothesis, there exists a 0 Proposition 1: The domatic number of a line graph of vertex domatic partition P of with

Ok where Ok is an odd graph is equal to 21k . classes. Therefore PVLG 0 is a vertex Proof: The edge domatic number of a graph is evidently equal to the domatic number of its line graph domatic partition of with k0 classes and it [1]. The degree of each vertex of line graph of is implies that d( L ( G )) ( G ) . 22k and this implies that its domatic number is at For equality, we have the following Cases. most . Therefore d L Ok 21 k . If G is a cycle C p with p is divisible by 3. Then d(L(G)) (G) 1. Theorem 9: For any connected graph G , c If is a cycle with is not divisible by 3. (G ) d ( L ( G )) ( G ) 1, where d L G is e Then d(L(G)) (G) . the domatic number of LG , G is the minimum degree of a vertex of G and (G) is the minimum degree Finally, we give the following e Characterization. of an edge of G . Theorem 10: Let T be a tree, let LT be the Proof: The number (G) is equal to the minimum e minimum degree of a vertex of LT(). Then degree of an edge of the line-graph of G . According to d( L ( T )) ( L ( T )) 1. [1], the domatic number of this line graph cannot be Proof: Let us have the colors1,2,..., (LT ( )) 1; we greater than (G) 1. e shall color the vertices of . First we choose a Now we shall prove the first inequality terminal vertex v0 of and color it by 1. Now let (G) d(L(G)) by the method of induction. If the us have a vertex v of with end verticesuw, ; degree of each vertex of LG is greater than or equal suppose that all vertices incident with w are already to k , where k is an arbitrary positive integer, then colored. Moreover, if the number of these vertices is there exists a domatic partition of with less than (L(T)) 1, we suppose that they are k classes. For k 1 the result is true. The required colored by pair wise different colors. In the opposite partition consists of one class equal to the whole case we suppose that all colors occur among the colors of these

IJSER © 2012 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 5 ISSN 2229-5518 vertices. Now we shall color the vertices adjacent to with (L(T)) 1classes and u and distinct from v . We color them in the following d( L ( T )) ( L ( T )) 1 . According to Theorem 9, it way. If there are colors by which no vertex adjacent to cannot be greater, therefore d(L(T)) (L(T)) 1. v is colored, we use all of them. (This must be always possible according to the assumption). If the number of vertices to be colored is less than (L(T)) 1, we REFERENCES: color them by pair wise distinct colors; in the opposite [1] F. Harary, Graph Theory, Adison Wesley, Reading case we color them by using all the Mass (1969). colors1,2,..., (LT ( )) 1 (it may form the color [2] T.W.Haynes, S.T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, Marcel Dekker, class). The result is a coloring of vertices of LT()by Inc., New York (1998). the colors with the property that [3] T.W.Haynes, S.T. Hedetniemi and P. J. Slater, each vertex is adjacent to vertices of all colors different Domination in graphs, advanced topics, Marcel Dekker, Inc., New York (1999). from its own. If Ai for i1,2,..., ( L ( T )) 1, is the [4] E.J.Cockayne and S.T. Hedetniemi, Towards a theory of set of all vertices of T colored by i , then the sets domination in graphs, Networks 7, (1977), 247 – 261.

AAA12, ,..., LT 1 form a domatic partition of