A graph G is said to be domination subdivision stable ( DSS ) if g ( Gsd uv ) = g ( G ), for all u, v ÃŽ V ( G ), u adjacent to v. In this paper we have provided two methods of obtaining a DSS graph from a non DSS graph

K. KARTHIKA

M. YAMUNA

English

Innovare Academic Sciences: E-Journals

 et al. Innovare Journal of Education, Vol 1, Issue 1, 2013,      Original Article  GENERATING DSS GRAPH BY EDGE SUBDIVISION AND EDGE CONTRACTION M. YAMUNA1, K. KARTHIKA2 SAS, VIT University, Vellore, Tamilnadu, India ,Email: karthika.k@vit.ac.in Received: 14 May 2014, Revised and Accepted:17 June 2014 ABSTRACT A graph G is said to be domination subdivision stable ( DSS ) if  ( Gsd uv ) =  ( G ), for all u, v  V ( G ), u adjacent to v. In this paper we have provided two methods of obtaining a DSS graph from a non DSS graph. Keywords: domination, subdivision, contraction.  INTRODUCTION A set of vertices D in a graph G = ( V, E ) is a dominating set if every vertex of V – D is adjacent to some vertex of D. The cardinality of the smallest dominating set of G is called the domination number of G and it is denoted by γ ( G ). A vertex in V – D is k – dominated if it is dominated by at least k – vertices in D. For properties related to graph theory we refer to [1] The open neighborhood of vertex v  V ( G ) is defined by N ( v ) = { u  V ( G ) | ( u v )  E ( G ) } while its closed neighborhood is the set N [ v ] = N ( v )  { v }. The private neighborhood of v  D is defined by pn [ v, D ] = N ( v ) – N ( D – { v } ). We indicate that u is adjacent to v by writing u  v. For properties related to domination we refer to [2]. MATERIALS AND METHODS An elementary edge contraction of a graph G is obtained by removal of u and v and the addition of a new point w adjacent to those points which u or v was adjacent. G uv is the graph obtained by contracting uv. In [3], Tamara Burton et al. defined a graph to be domination dot critical ( DDC ) if  ( Guv ) <  ( G ),  u, v  V ( G ). They have proved the following result. The subdivision of some edge e with endpoints { u, v } yields a graph containing one new vertex w, and with an edge set replacing e by two new edges, { u, w } and { w, v }. We shall denote the graph obtained by subdividing any edge uv of a graph G, by Gsd uv. Let w be a vertex introduced by subdividing uv. We shall denote this by Gsd uv = w. In [4], M. Yamuna et al have introduced the concept of domination subdivision stable graphs. A graph G is said to be domination subdivision stable ( DSS ) if the  - value of G does not change by subdividing any edge of G.  Example  Fig. 1:  ( G ) =  ( Gsd uv) = 2. This is true for all e = ab  E ( G ). Here G is a DSS graph. In [4], they have proved the following result R1. A graph G is DSS if and only if for every u, v ∈ V ( G ), either there is a γ - set containing u and v or, there is a γ - set D such that 1. pn[ u, D ] = { v } or 2. v is 2 – dominated. R2. Every DDC graph is DSS. RESULTS AND DISCUSSIONS Theorem 1  Let G be a graph which is not DSS. There is a subdivision graph of G which is DSS. Proof  Since G is not DSS, there is at least one u, v  V ( G ), u  v such that γ ( Gsd uv ) ≠ γ ( G ). By [ R1 ], either both u, v  D or pn[ u, D ]  2, v  pn[ u, D ], where D is any γ – set for G. Let E = { es / es = ( us, vs ), us  vs, γ ( Gsd usvs ) ≠ γ ( G ) }.  Let Xi = N ( vi ) – { ui }, Yi = N ( Xi ), Zi = N ( ui ) – { vi }. For any em  E ( G ), let γ ( Gsd umvm ) = wm. Dm = D  { wm } is a γ – set for Gsd umvm. Let Gm = Gsd umvm. Case 1 em  E such that um  D, vm  D, m = 1, 2, …, k. For all xm  Xm, γ ( Gm sd vmxm) = γ( Gm ), since Dm – { wm }  { vm } is a γ – set for Gm such that every xm Xm, xm  D is 2 – dominated. For all ym  Ym, such that ym  D, γ ( Gm sd xmym) = γ( Gm ), since Dm – { wm }  { xm } is a γ – set for Gm for every xm Xm. Also γ ( Gm sd umwm ) = γ ( Gm sd wmvm ) = γ ( Gm ), since pn[ wm, D ] = vm. Case 2 em  E such that um, vm  D, m = 1, 2, …, p. Note that k + p = | E |. For all xm  Xm, γ ( Gm sd vmxm) = γ( Gm ), since Dm – { wm }  { vm } is a γ – set for Gm such that every xm Xm, xm  D is 2 – dominated. Similarly for all zm  Zm, γ ( Gm sd umzm) = γ( Gm ), since Dm – { wm }  { um } is a γ – set for Gm such that every zm  Zm, zm  D is 2 – dominated. Also since wm is selfish, γ ( Gm sd umwm ) = γ ( Gm sd wmvm ) = γ ( Gm ). From case 1, for all em  E ( G ) in graph Gm, for all xm  Xm, ym  Ym such that γ ( Gsd vmxm ) ≠ γ ( G ), γ ( Gsd xmym ) ≠ γ ( G ) become DSS in Gm. Vol 2, Issue 3 , 2014                ISSN: 2347-1573 Karthika et al. Innovare Journal Of Engg. & Tech. 1-2 2  Similarly from case 2, for all em  E ( G ) in a graph Gm, for all xm  Xm, zm  Zm such that γ ( Gsd vmxm ) ≠ γ ( G ) and γ ( Gsd umzm ) ≠ γ ( G ) become DSS in Gm. Also the new edges introduced are DSS and the edges which were subdivision stable in G are DSS in Gm also. Let Em = { em / em = ( um vm ), um  vm, γ ( Gsd umvm ) ≠ γ ( Gm ) }, clearly | Em | < | E |. If Gm is DSS we terminate here else starting from Em we repeat the same procedure to obtain a new graph Gm + 1. If Gm + 1 is DSS we terminate here else we continue to generate a sequence of graphs Gm, Gm + 1, … such that 1. | Em + 1 | < | Em |. 2. γ ( Gm + 1 ) = γ ( Gm ) + 1. until we obtain a graph that is DSS. Theorem 2 Let G be a graph which is not DSS. There is a graph H of G which is DSS, if H can be obtain from G by a sequence of elementary contraction. Proof  Since G is not DSS there is at least one u, v  V ( G ), u  v such that γ ( Gsd uv ) ≠ γ ( G ). By [ R1 ], either both u, v  D or pn[ u, D ]  2, v  pn[ u, D ], where D is any γ – set for G. Let E = { es} and | E | = k + p. Let Xi = N ( vi ) - { ui }, Yi = N ( Xi ), Zi = N ( ui ) – { vi }. For any em  E ( G ), let γ ( G umvm ) = γ ( G ), by [ R2 ]. D is a γ – set for G umvm also. Let Gm = G umvm. Case 1 em  E such that um  D, vm  D, where m = 1, 2, …, k.  For all xm  Xm, γ ( Gm sd ( umvm ) xm ) = γ ( Gm ), since xm is 2 – dominated in Gm. For all ym  Ym such that ym  Dm, γ ( Gm sd xmym ) = γ ( Gm ), since xm is 2 – dominated in Gm. Case 2 em  E such that um, vm  D, where m = 1, 2, …, p. Note that k + p = | E |. For all xm  Xm, such that xm  Dm. γ ( Gm sd ( umvm ) xm ) = γ ( Gm ), since umvm is 2 – dominated in Gm.  Similarly for all zm  Zm such that zm  Dm, γ ( Gm sd ( umvm ) zm ) = γ ( Gm ), since umvm is 2 – dominated in Gm. The edges which are DSS in G are DSS in Gm also. Let Em = { em / em = ( um vm ), um  vm, γ ( Gm sd umvm) ≠ γ ( G ) }, clearly | Em | < | E |. If Gm is DSS, we terminate here else starting from Em we repeat the same procedure to obtain a new graph Gm + 1. If Gm + 1 is DSS we terminate here else we continue to generate a sequence of graphs Gm, Gm + 1, Gm + 2, …, Gq such that 1. | Em + 1 | < | Em |. 2. γ ( Gm + 1 ) = γ ( Gm ). until we obtain a graph Gq + 1 such that γ ( Gq + 1 uv ) < γ ( Gq + 1), for all u, v  V ( Gq + 1 ), u  v, that is till we obtain a dot critical graph Gq. We know that every dot critical graph is DSS. This implies Gq is DSS.  CONCLUSION  The paper provides two methods of obtaining DSS graph from a non DSS graph by applying graph operations. By applying other graph operations like complement, dual, cross product we can generate DSS graphs from non DSS graphs.  REFERENCES 1. Harary. Graph Theory. Addison Wesley. Narosa Publishing House; 2001.  2. Burton T and Sumner D. Domination Dot – Critical Graphs. Discrete Math., 2006; 306: 11 – 18. 3. Haynes TW, Hedetniemi ST and Slater PJ. Fundamentals of Domination in Graphs. Marcel Dekker. New York; 1998. 4. Yamuna M and Karthika K. Domination Subdivision Stable Graphs. International Journal of Mathematical Archive. 2012; 3(4):1467-1471. 

GENERATING DSS GRAPH BY EDGE SUBDIVISION AND EDGE CONTRACTION

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GENERATING DSS GRAPH BY EDGE SUBDIVISION AND EDGE CONTRACTION

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