The Comparison of Neighbourhood Set and Degrees of an Interval Graph G Using an Algorithm

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1 The Comparion of Neighbourhood Set and Degree of an Interval Graph G Uing an Algorithm Dr.A.Sudhakaraiah, K.Narayana Aitant Profeor, Department of Mathematic, S.V. Univerity, Andhra Pradeh, India Reearch Scholar, Department of Mathematic, S.V. Univerity, Andhra Pradeh, India ABSTRACT: In general, any mathematical object involving point and connection among them can be called a graph or an interval graph or hyper graph. For a great diverity of problem uch network and pictorial repreentation may lead to a olution. In thi domination i a rapidly developing area of reearch in graph theory, and it variou application to ad,hoc network, ditributed computing, ocial network and web graph partly explain the increaed interet. In thi paper focue on neighbourhood et and the main aim of the comparion of neighbourhood et and degree of an interval graph G uing an algorithm. KEYWORDS: Interval family, Interval graph, Neighbourhood Number, Degree. I. INTRODUCTION Graph are very convenient tool for repreenting the relationhip among object, which are repreented by vertice. In their turn, relationhip among vertice are repreented by connection. Example of uch application include databae, phyical network, organic molecule, map coloring ignal flow graph etc. At preent graph theory i a dynamic field in both theory and application. In thi ection, the concept of domination, minimum neighbourhood et of an interval graph[]. Let G (V, E) be a graph. The neighborhood et of a vertex V in G[,,] i defined a the et of vertice adjacent with V (including V) and i denoted by nbd[v]. A et S of vertice in G i called a neighborhood et of G if G= G[ nbd[ v]], where G nbd[v] i the vertex induced ub graph of G. The V S neighbourhood number of G i defined a the minimum cardinality of a neighbourhood et S of G[]. The degree of a vertex V in an interval graph G i the number of edge of G incident with V and it i denoted by degree of V that i deg(v). The maximum or the minimum degree among the vertice of G i denoted by (G) orδ(g). In thi connection we will conider the maximum degree of vertice v from G correponding to I In thi paper we dicu an algorithm for finding a minimum neighbourhood et, maximum degree of an interval graph correponding to an interval family[,,,,]. In thi chapter we dicu the comparion of minimum neighbourhood et and maximum vertex degree. In thi chapter we propoe an algorithm for the comparion of minimum neighbourhood et and maximum degree of an interval graph correponding to an interval family I[]. II. PRELIMINARIES Let I= {,,., n}be an interval family. Each interval i in I i repreented by [a, b ] for i=,,...,n here a i called the left end point labeling and b i the right end point labelling of the interval i. Without lo of generality, we may aume there are n endpoint which are ditinct. The interval are labelled in the increaing order of their end point. Two interval i and j are aid to interect each other if they have non- empty interection. A graph Copyright to IJIRSET DOI:./IJIRSET

2 G(V,E)i called an interval graph if there i a one to one correpondence between V and I uch that two vertice of G are joined by an edge in E if and only if there correponding interval in I interect that i if i= [a, b ] and j= a, b, then i and j interect mean either a b or a b []. Next we aume that the label of the interval atify uch an ordering of n interval. We proceed in the following manner to find a minimum neighbourhood et and maximum degree of an interval graph toward an algorithm. Let min (i) and max (i) denote the larget interval in nbd[i]. Let next[i] = j iff b a and there do not exit k uch thatb a a. If there i no uch j, then define next (i) = null. We tart S= {max ()} LI denote the larget interval in S. Theorem: For any interval graph G of order N correponding to an interval family I, Then MNS N. Proof: In thi theorem it will arie the following cae.. To find the vertex degree of G of order n correponding to an interval family I.. To find neighbourhood et of S correponding to I.. To find the minimum neighbourhood et of G correponding to an interval family I, Cae(i) :To find vertex degree: Let I={I, I,.., I } be an interval family and let G be an interval graph correponding to an interval family I. We will find the maximum degree of vertex from an interval graph G correponding to I. The degree of a vertex V i an interval graph G i the number of edge of G incident with V and it i denoted by degree of V i.e., deg (V). The maximum or minimum degree among the vertice of G i denoted by (G)or δ(g). in thi connection we will conider the maximum degree of vertex V from G correponding to I. Generally we call that a vertex i called odd or even depending on whether degree i odd or even. If deg (V) =0, then V i called an iolated vertex of G. If deg (V) =, then V i called an end vertex or a pendent vertex of G and an edge incident with an end vertex i called an end edge or a pendent edge. The degree of an edge u, v i defined to be degree u+ degree of v-. The maximum or, minimum degree among the edge of G denoted by (G)or δ(g). an edge i called an iolated edge if deg (uv)=0. Then the graph mut be diconnected but we conider only connected graph G correponding to I. Cae(ii) :To find neighbourhood et of S correponding to I: Again we will find the minimum neighbourhood et of an interval graph G correponding to an interval family I the neighbourhood of a vertex V in G i defined a the et of vertice adjacent with V including V and i denoted by nbd(v). A et S of vertice in G i called a neighbourhood et of G if G == G[ nbd[ v]],, where G nbd[v] i V S the vertex induced ub graph of G. The neighbourhood number of G i defined a the minimum cardinality of neighbourhood et S(G). We have to find the minimum neighbourhood et toward an algorithm of an interval graph correponding to an interval family I. Let S= {I,I,...I k } where I =max(), I k =max(min(next(i k- ))) where k be the et contructed by the algorithm. Let the interval be i and j are any two interval I uch that i<j<max(i) then j interect max(i), all the interval from to max() belonging to neighbourhood of max() and alo the ubet {j,...k(i) i contained in nbd(j) nbd(k)} i.e., any interval between j and k i covered by either j or k. i.e., all the interval j uch that i <j<i + are contained in nbd(i ) nbd(i + ), for all k. Where i + =max (min (next (i ))). By the algorithm next (i k ) =null. Therefore the interval from i to n belong to nbd (i k ). So all the interval in I belong to nbd [i ] nbd [i ]... nbd [i k ]. And I= nbd(i ) nbd(i )... nbd(i k ).further an induced ub graph of G on {i,...,i + } i contained in G[nbd[i ]]UG[nbd[i + ]].Hence G{ i,.., i } G[ nbd[ i ]]. But clearly G G{ i..., i }. { i,.., i } I i { i,.., i } I, Copyright to IJIRSET DOI:./IJIRSET.0.00

3 Cae(iii): To find the minimum neighbourhood et of G correponding to an interval family I : Let I = {i,j,...} be an interval family and G i an interval graph correponding to an interval family I. Where I = max (), I k =max (min (next (i k- ))) where k. the right end point of interval in S are ordered uch that b < b <...b. Let j and k be two interval in I uch that k= max (min (next (j))). We conider if x i any interval uch that b x > b, than it doe not interect min (next (i )).thi implie that the edge [min (next (i )), next (i )] doe not belong to nbd(t). Since NS i a neighbourhood et of I, there exit an interval u NS uch that b and the Edge[min(next(i )),next(i )] G[nbd(u). that there exit an interval u NS uch that b b and the Edge[min(next(i )),next(i )] G[nbd(u)].that i there exit an interval u NS uch that i < u i and the Edge[min(next(i )),next(i )] G[nbd(u)].thu inbetween any two conecutive interval in, we get an interval u NS,uch that i < u < i,where m therefore beide i, there are atleat m- ubinterval in NS. Hence NS m =. III. TO FIND MINIMUM NEIGHBOURHOOD SET OF AN INTERVAL GRAPH USING AN ALGORITHM Input: interval family I= {,,...,n} Output: Minimum neighbhourhood et of the interval graph G (I) Step: et S= {max ()}. Step : LI=the larget interval in S. Step : compute next (LI). Step : if next (LI) = null, then go to tep. Step : find max (min (next (LI))). Step : S=SU max (min(next(li))),go to tep. Step : END IV. Experimental problem : for any interval graph G of order n, MNS N. Fig : Interval Family I Copyright to IJIRSET DOI:./IJIRSET.0.00

4 V. DEGREE OF VERTICES deg () =, deg () =, deg () =, deg () = deg () =, deg () =, deg () =, deg ()= deg () =, deg () =, deg () = Since = deg()= deg()= Here n=, =, δ =, MNS = {,,}= Therefore the reult i MNS N. Fig : Interval graph G The reult i true. Therefore we conider the maximum degree of g i either deg() or deg() i. VI. TO FIND MINIMUM NEIGHBOURHOOD SET (MNS) OF AN INTERVAL GRAPH CORRESPONDING TO AN N INTERVAL FAMILY I nbd()= {,,}, min()=, max()= & next()= nbd()= {,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= Copyright to IJIRSET DOI:./IJIRSET.0.00

5 nbd()= {,,,}, min()=, max()= & next()= nbd()= {,,,,, }, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,}, min()=, max()= & next()= nbd()= {,,,,,}, min()=, max()= & next()= null nbd()= {,,,}, min()=, max()= & next()= null nbd()= {,,}, min()=, max()=& next()= null VII. PROCEDURE FOR FINDING A MINIMUM NEIGHBOURHOOD SET(MNS) OF AN INTERVAL GRAPH USING AN ALGORITHM INPUT: INTERVAL FAMILY I= {,,,} STEP: S= {max ()} = {} STEP: LI= the larget interval in S= STEP: Next (LI) = Next () = Step:max(min(next(LI)))= max(min(next()))= max(min())= max()= STEP: S= S {} = {,}, go to tep STEP: LI= the larget interval in S= STEP: Next (LI)= Next()= Step:max(min(next(LI)))= max(min(next()))= max(min())= max()= STEP: S= S {} = {,,}, go to tep STEP: LI= the larget interval in S= STEP: Next (LI) = Next () = null, go to tep STEP: End. OUTPUT: S= {,, } i the minimum neighbourhood et of the interval graph G. VIII. EXPERIMENTAL PROBLEM FOR ANY INTERVAL GRAPH G OF ORDER N CORRESPONDING TO AN INTERFAMILY, MNS (n + ) (δ ) δ Fig : Interval family I Copyright to IJIRSET DOI:./IJIRSET.0.00

6 Fig : Interval graph G deg()=, deg()=, deg()=, deg()=, deg()=, deg()=, deg()=, deg()=, deg()=, deg()=, deg()=, deg()= Here =, δ =, n =, MNS = {,,}= Therefore the reult i MNS (n + ) (δ ) δ Therefore the reult i true. Since = deg() = IX. DEGREE OF VERTICES. X. NEIGHBOURHOOD SET OF VERTICES nbd()= {,,}, min()=, max()= & next()= nbd()= {,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()=& next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,}, min()=, max()= & next()= nbd()= {,,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= null Copyright to IJIRSET DOI:./IJIRSET.0.00

7 nbd()= {,,,}, min()=, max()=& next()= null nbd()= {,,}, min()=, max()=& next()= null XI. PROCEDURE FOR FINDING A MINIMUM NEIGHBOURHOOD SET(MNS) OF AN INTERVAL GRAPH G USING AN ALGORITHM INPUT: INTERVAL FAMILY I= {,,,} STEP: S= {max ()} = {} STEP: LI= the larget interval in S = STEP: Next (LI) = Next () = Step: max(min(next(li))) = max(min(next())) = max(min()) = max()= STEP: S= S {} = {,}, go to tep STEP: LI= the larget interval in S= STEP: Next (LI) = Next () = Step:max(min(next(LI)))= max(min(next()))= max(min())= max()= STEP: S= S {} = {,,}, go to tep STEP: LI= the larget interval in S= STEP: Next (LI) = Next ()= null, go to tep STEP: End. OUTPUT: S= {,, } i the minimum neighbourhood et of the interval graph G. XII. EXPERIMENTAL PROBLEM FOR ANY INTERVAL GRAPH G OF ORDER N CORRESPONDING TO N INTERFAMILY, MNS [(n + ) δ] Fig : Interval FamilyI Copyright to IJIRSET DOI:./IJIRSET.0.00

8 Fig : Interval Graph G XIII. DEGREE POF VERTICES deg()=, deg()=, deg()=, deg()= deg()=, deg()=, deg()=, deg()= deg()=, deg()=, deg()=, deg()= Here =, δ =, n =, MNS = {,,,}= Therefore the reult i MNS [(n + ) δ] Therefore the reult i true. Since = deg() or deg () or deg () or deg () or deg () or deg () or deg () or deg () = XIV. NEIGHBOURHOOD OF VERTICES FROM G nbd()= {,,}, min()=, max()= & next()= nbd()= {,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= Copyright to IJIRSET DOI:./IJIRSET.0.00

9 nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= nbd()= {,,,,}, min()=, max()= & next()= null nbd()= {,,,}, min()=, max()=& next()= null nbd()= {,,}, min()=, max()=& next()= null XV. PROCEDURE FOR FINDING A MINIMUM NEIGHBOURHOOD SET(MNS) OF AN INTERVAL GRAPH USING AN ALGORITHM INPUT: INTERVAL FAMILY I = {,,,}. STEP: S= {max ()} = {} STEP: LI= the larget interval in S= STEP: Next (LI) = Next () = Step:max(min(next(LI)))= max(min(next()))= max(min())= max()= STEP: S= S {} = {,}, go to tep STEP: LI= the larget interval in S= STEP: Next (LI) = Next () = Step:max(min(next(LI)))= max(min(next()))= max(min())= max()= STEP: S= S {} = {,,}, go to tep STEP: LI= the larget interval in S= STEP: Next (LI) = Next () = Step:max(min(next(LI)))= max(min(next()))= max(min())= max()= STEP: S= S {} = {,,,}, go to tep STEP: LI= the larget interval in S= STEP: Next (LI) = Next () = null STEP: End. OUTPUT: S= {,,, } i the minimum neighbourhood et of the interval graph G. XVI. CONCLUSION In thi paper focue on neighbourhood et and the main aim of the comparion of neighbourhood et and degree of an interval graph G correponding to an interval family I uing an algorithm. In future effort in the paper eventually open up many an avenue in the field of reearch on interval graph. REFERENCES [] E. Sampath Kumar, P.S. Neeralagi, The neighbour of graph, Indian J. Pure.Appl.Math.,,, -. [] Kulli, V.R., Sigarkanti, S.K., Further reult on a neighbourhood number of a graph, Indian, J. Pure. Appl. Math.,,,-. [] Even, S., Graph Algorithm, Computer Science Pre, Rockville, MD, 0. [] E.J.Cockayne, S.T.Hedetniemi, Toward a theory of domination in graph, Network, Vol.(), -. [] Bertoi, A.A., Total domination in Interval graph, Inform. Proce., Lett.,,,-. [] E. Horowitz and S. Sahani; Fundamental of computer algorithm. Galgotia Publication, New Delhi() [] B. Mahewari, Y. Lakhmi naidu, L. Nagamuni reddy and A. Sudhakaraiah, Minimum Global dominating et of an interval graph, International J. of Math. Sci. & Appl., Vol., No. IV( July, 0), pp. -. [] B. Mahewari, Y. Lakhmi naidu, L. Nagamuni reddy and A. Sudhakaraiah, A polynomial time algorithm for finding a minimum independent neighborhood et of an interval graph, Graph theory note of newyork XLVI, -(00) New York academy Science. [] Gupta, U.I., Lee, D.T. Leung, J.Y.T. Efficient algorithm for interval graph and circular-arc graph, Network,,, -. []Keil,J.M., Total domination in interval graph, Inform. Proce. Lett.,,, -. []Bertoi, A. A. Bonucelli, M.A. Some parallel algorithm on interval graph, Dicrete. Appl. Math.,,, -. []Even, S. Graph algorithm, Computer Science Pre, Rockville, MD, 0. Copyright to IJIRSET DOI:./IJIRSET.0.00