Metric Dimension of Some Graphs under Join Operation

Transcription

1 Global Joural of Pure ad Applied Matheatics ISSN Volue 3, Nuber 7 (07), pp Research Idia Publicatios Metric Diesio of Soe Graphs uder Joi Operatio B S Rawat ad P Pradha** Departet of Matheatics ad Statistics Gurukula Kagri Uiversity, Haridwar (UK), Idia Abstract I this paper, we fid the etric diesio for oi of two paths ad geeralized the Caser s result for oi of a path ad a coplete graph We have also iproved the results give by Shahida, A T ad M S Suitha Throughout the paper, fiite ad siple graphs have bee cosidered Keywords: Resolvig set, Basis, Metric Diesio INTRODUCTION Navigatio ca be studied i a graph structure fraework i which the avigatio aget oves fro ode to ode of a graph space The robot ca locate itself by the presece of distictly labeled ladark odes i a graph space If the robot kows its distaces to a sufficietly large set of ladarks, its positio o the graph is uiquely deteried This suggests the followig proble: give a graph, what are the fewest uber of ladarks eeded, ad where they should be located, so that the distace to the ladarks uiquely deteries the robot s positio o the graph? A iiu set of ladarks which uiquely deteries the robot s positio is called basis, ad the iiu uber of ladarks is called the etric diesio of graph Motivated by this proble, the cocept of etric diesio was itroduced by Slater [7] ad idepedetly by Harary ad Melter [] Correspodig Author Eail- bipedrarawat80@gailco (B S Rawat) **Eail- ppradha4@gailco (P Pradha)

2 333 B S Rawat ad P Pradha Let W { w, w, w3,, w k } be a ordered subset of VG; ( ) the the etric represetatio of v V ( G) with respect tow is defied as the k tuple r v W d v w d v w d v w3 d v w k ( / ) { (, ), (, ), (, ),, (, )} The set W is called a resolvig set of G if for all u vad u, v V( G) satisfy r( v / W) r( u / W) A resolvig set W of G with the iiu cardiality is the basis of G The uber of eleets i basis is called etric diesio of G ad is deoted by ( G), thus ( G) i{ W : W is a resolvig set of G} Khuller et al [9] studied the etric diesio otivated by the robot avigatio while Chartrad et al [] characterizes all the graphs of order havig etric diesio Caceres et al [4] have deteried the etric diesio for Cartesia product of graphs Saputro et al [0] have show etric diesio of cob product of graphs Buczkowski et al [8] deteried the etric diesio of wheelw K C, for 3 Caceres et al [5] the etric diesio of fa f K P, for ad Toescu ad Javaid [3] the etric diesio of Jahagir graph J, for Motivated by [3], [4], [5], [6], [8], [0] ad []; we have obtaied soe results uder the oi operatio of two graphs JOIN OF TWO GRAPHS The oi of two graphs G ad G, deoted by G G ; is a graph with vertex set V ( G) V( G) ad the edge set E( G ) E( G ) { uv u V( G ), v V( G )}I the graph G G each vertex of G is adacet to the vertices of G ad vise versa ie d( u, v ) ; u V( G ), v V( G ) i i Khuller et al [9] have derived the followig results: : Metric diesio of a graph G is if ad oly if G is a path : If G ( V, E) is a graph with etric diesio ad { a, b} V( G) is a basis for G, the (i) There is a uique shortest path betwee a ad b (ii) The degrees of vertices a ad b are at ost three (iii) Other vertices lyig o uique shortest path betwee a ad b have degree at ost five

3 Metric Diesio of Soe Graphs uder Joi Operatio MAIN RESULT O the basis of above results, we have obtaied ecessary coditio for the basis of oi of two graphs: Lea 3: If VG ( ) ad VG ( ) are vertex sets of two o ull graphs G ad G respectively ad W is a basis of graph G, G the W V( G ), i, Proof: Let V( G ) { u, u, u3, u}; V( G ) { v, v, v3, v}; be the vertex sets of graphs G ad G respectively ad let W be the basis for the graphg G If W V ( G ), the we have r( v / W) r( vk / W) (,,,,); k ; v, vk V( G ) ad it gives a cotradictio Siilarly, ifw V ( G ), the r( u / W) r( uk / W) (,,,,); k ; u, uk V( G ) ad a cotradictio agai So W ca t be a basis of G G Thus W ust cotai at least oe vertex of each graph i I the ext lea we have obtaied the lower boud for etric diesio of oi of two paths Lea 3: If P ad P ;, are two paths, the ( P P ) 3 Proof: Let P ad P be two paths ad W be a basis of P followig possible cases arise: P The there are Case I: If, obviously P P K 4 ad therefore ( P P) 3 Case II: If ad, clearly ( P P ) Now we assue that ( P P ), the W { ui, v } ; ui V ( P ) ad v V ( P); fors a basis of P P Degree of each u i i this case is at least 4 i P P ; which is cotradictio of result (ii), therefore W cotaiig oly two eleets caot be a basis for P P Case III: If ad The clearly degree of each vertex of P P is at least 4, which is agai a cotradictio of result (ii) Thus we coclude that o two vertices of P P for a basis for P P ie ( P P) 3 I the followig Fig, Fig ad Fig 3, we have show all the above three cases

4 3334 B S Rawat ad P Pradha I the followig theore we have obtaied sufficiet coditio for basis of oi of two paths Theore 33: Let W V ( P ) ad W V ( P ) be two ordered sets of graph P P such that

5 Metric Diesio of Soe Graphs uder Joi Operatio 3335 (i) r ( ui / W) r( u / W); i ad ui, u V( P ) i cotext of graph P P (ii) r ( vs / W) r( vt / W); s t ad vs, vt V( P ) icotext of graph P P (iii) For each i ad s ; both r ( ui / W ) ad r ( vs / W ) should ot as (,,,,) at the sae tie ie if r ( ui / W ) (,,,,), the r ( v / W ) (,,,,) ad vice-versa s The the ordered set W W W is a resolvig set for the graph P P ad if both W ad W are sallest sets also; the W is a basis for the graph P P Proof: Let us take W W W { u, u, u3, ul, v, v, v3, vk} such that r( u / W ) r( u / W ),i P P i i The r( u / W) r( u / W), i () i Siilarly r( v / W) r( v / W), s t () s t Now r( u / W) r( v / W) is possible oly whe, i r( u / W) r( v / W) (,,,,) i s s But fro coditio (iii) of the theore both r ( ui / W ) ad r ( vs / W ) should ot as (,,,,) at the sae tie Therefore The by (),() & (3), we coclude that r ( u / W) r ( v / W); i ad s (3) (3) i r( w / W) r( w / W), w, w V ( P P ) i i i s Thus W i e W W is a resolvig set for P P If W ad W are sallest the obviously W WW is also a sallest set Hece W W W is a sallest resolvig set ie a basis for the graph P P Exaple 34: Let P 4 ad P 6 be two paths, the cosider two ordered subsets W { u, u} ad W { v, v4} of P 4 ad P 6 respectively r( u / W ) (,) i cotext of graph P P Now r( u / W ) (,)i cotext of graph P P r( v / W ) (,)i cotext of graph P P 4 6

6 3336 B S Rawat ad P Pradha r( v / W ) (,) i cotext of graph P P r( v / W ) (,) i cotext of graph P P r( v / W ) (,)i cotext of graph P P Here we fid that (i) r ui W r u W i ui u V P4 P4 P6 ( / ) ( / ); ad, ( ) i cotext of graph (ii) r vs W r vt W s t vs vt V P6 P4 P6 ( / ) ( / ); ad, ( ) icotext of graph (iii) r ( ui / W ) (,,); i 4 icotext of graph P4 P6 The by theore 33; W W { u, u, v, v4} is a resolvig set for P 4 P 6 Sice both W ad Ware sallest sets; therefore W W { u, u, v, v4} is a basis for the graph P 4 P 6 ad ( P4 P6) 4 Theore 35: Let P ad P be two paths, the etric diesio of P P is 3; 5ad 3, 4; 5ad 6 or 4 6 ad 4 5, 5; 6 ; 3 ad 7, ( P P) ; 4 6 ad 7, ; 7 ad 7, Proof: Let { ui ui V( P )} ad { vi vi V ( P )} be the vertex sets of paths P ad P respectively The Case I: If 5ad 3 (a) If 5ad 3 Suppose W { v, u, u } V( P P ), we show that W is a resolvig set for P P For this we take the represetatio of ay vertex of V ( P P ) \ W with respect tow :

7 Metric Diesio of Soe Graphs uder Joi Operatio 3337 r( v / W ) (,,) r( v / W ) (,,) 3 r( u / W ) (,,) Sice these represetatios are pair wise distict it follows that ( P P) 3 But ( P P) 3Therefore ( P P) 3 ; whe 5ad 3 (b) If ad 3, we cosider the set Now the represetatios of V ( P P ) \ W with respect tow : r( v / W ) (,,) r( v / W ) (,,) 3 Proceedig i sae way as above, we observe that there are o two vertices havig the sae etric represetatios with respect to W, iplyig that ( P P) 3 ad ( P P ) 3 So ( P P) 3, for ad 3 Case II: If 5ad 6 or 4 6ad 4 5, the (a)if 5ad 6, Cosider the set W { v, v4, u, u} V( P P ) Now the represetatios of V ( P P ) \ W with respect to W : r( v / W ) (,,,) r( v / W ) (,,,) 3 r( v / W ) (,,,) 5 r( v / W ) (,,,) 6 r( u / W ) (,,, ) 3 r( u / W ) (,,, ) r( u / W ) (,,,) We fid that there are o two vertices havig sae represetatios iplyig that ( P P ) 4 Now we show that ( P P) 4, by provig that there is o resolvig set W with cardiality 3 Now followig sub cases arise:

8 3338 B S Rawat ad P Pradha (i) If W cotais oe vertex of observe that P ad two vertices of P ie W { u, v, v }, we i k (a) For ad k 4 or 3 ad k 5differet vertices of P have differet represetatios with respect to W but i this case r( v / W) (,,); k () ad r( u / W) (,,); u N( u ) () l l i Therefore r( v / W) r( ul / W); thus a cotradictio (b) If ad k 4 or 3 ad k 5; the there exist at least two vertices of V ( P P ) \ W havig the sae etric represetatio with respect to W, a cotradictio agai Therefore cardiality of W ca t be three i this case ie ( P P) 4 So ( P P ) 4 (ii) Siilarly if W cotais two vertices of cotradictio P ad oe vertex of P ; we agai get a Case II (b): If 4 6ad 4 5 SupposeW { v, v4, u3, u4} V( P P), the r( v / W ) (,,,) r( v / W ) (,,,) 3 r( v / W ) (,,,) 5 r( v / W ) (,,,) 6 r( u / W ) (,,,) r( u / W ) (,,,) r( u / W ) (,,,) 5 There are o two vertices havig the sae etric represetatios with respect to W, iplyig that ( P P ) 4 Now proceedig o the sae aer as above ( P P) 4, therefore ( P P ) 4; whe 4 6 ad 4 5 Case III: For 6; cosider the setw { v3, v4, v6, u3, u5}, the the etric represetatios of the vertices of V ( P P ) \ W with respect to W :

9 Metric Diesio of Soe Graphs uder Joi Operatio 3339 r( v / W ) (,,,,) r( v / W ) (,,,,) r( v / W ) (,,,,) 5 r( u / W ) (,,,, ) r( u / W ) (,,,, ) r( u / W ) (,,,,) 4 r( u / W ) (,,,,) 6 Differet vertices of V ( P P ) \ W have differet etric represetatios with respect to W, therefore ( P P) 5 (3) Now i order to prove that ( P P) 5; we show that there is o resolvig setw such that W 4 For this followig possible cases arises: (i) Whe W cotais oe vertex of P ad three vertices of P ie W { ui, v, vk, vl} the at least two vertices u, u V( P ) such that r( u / W) (,,,) r( u / W); a cotradictio (ii) Siilarly whew cotais oe vertex of P ad reaiig three vertices of P, the at least two vertices v, v V ( P ) such that r ( v / W ) r ( v / W ); agai a cotradictio (iii) Whe W cotais equal uber of vertices fro both W { ui, u, vk, vl}; the we observe that P ad P ie (a) For i ad 4 or i 3 ad 5 differet vertices of P have differet represetatios with respect tow ad r( ui / W) (,,,); i (4) Siilarly for k ad l 4 or k 3 ad l 5 differet vertices of P have differet represetatios with respect to W ad i this case r( vk / W) (,,,); k l (5) By (4) ad (5) we coclude that r( vi / W) r( uk / W); ad this is a cotradictio (b) For i ad 4 or i 3 ad 5; k ad l 4 or k 3 ad l 5; the there exist at least two vertices of V ( P P ) \ W havig the sae etric represetatio with respect to W, which is a cotradictio Hece W ca t be a resolvig set with W 4 i e ( P P ) 4 ad also

10 3340 B S Rawat ad P Pradha ( P P ) 3; 5 ad 3 so ( P P ) 3i this case Therefore ( P P) 5 (6) By (3) ad (6) ( P P ) 5;whe 6 Case IV: For 3 ad 7 (a) If is eve ad 7, i this we take W cosistig of a sigle vertex of P ad a ordered set W of vertices of P such that W { u} ad W { v3, v5,, v },we get r( u / W) () i the cotext of graph P P r( u3 / W) () i the cotext of graph P P Obviously, r( ui / W) r( u / W); i ; i the cotext of graph P P ad i, 3 Ad r( v / W) (,,,) i the cotext of graph P P r( v / W) (,,,) i the cotext of graph P P r( v4 / W) (,,,) i the cotext of graph P P So o, r( v / W ) (,,,) i the cotext of graph P P Here we fid that (i) r( vi / W) r( v / W) i the cotext of graph P P ; i (ii) r( vi / W ) (,,,,) i i the cotext of graph P P The by theore 33, W W is a resolvig set for P PTo ake W W iiu; both W ad W should be iiu separately Here we have to ake W iiu; sice W cotaiig oe eleet is iiu For akig W iiu; we follow the steps give below: (i) We oit ed vertices of P (ii) We select alteratig vertices as the eleets ofw Thus; for P ; W, 7 is iiu ad ( P P) (7)

11 Metric Diesio of Soe Graphs uder Joi Operatio 334 (b) If is odd, the siilarly W { u} is a sallest ordered set i P such that r( u / W ) r( u / W ) i the cotext of graph P P ; i ad i, 3 i Now let W { v3, v5, v7, v }; the r v W P P ( / ) (,,,) i the cotext of graph r v W P P ( / ) (,,,) i the cotext of graph r v W P P ( 4 / ) (,,,) i the cotext of graph So o fially r( v / W) (,,,) i the cotext of graph P P Here we see that (i) r( vi / W) r( v / W) i the cotext of graph P P i (ii) r( vi / W ) (,,,,) i i the cotext of graph P P Therefore W WW is a resolvig set for P P Now we have to ake W iiu; sice W cotaiig oe eleet is iiu For akig W iiu; we follow the steps give below: (i) We oit oe ed vertex of P (ii) We select alteratig vertices as the eleets of W Thus; for P ; W, 7 Therefore iiu W W i e ( P P) (8) By (7) ad (8) ( P P) ; 3 ad 7 Case V: For 4 6 ad 7; the (a) If is eve, the we take two ordered sets W ad W fro P ad P respectively such that W { u, u4} adw { v 3, v 5, v 7 v },the r u W P P ( / ) (,) i the cotext of graph

12 334 B S Rawat ad P Pradha r u W P P ( 3 / ) (,) i the cotext of graph r u W P P ( 5 / ) (,) i the cotext of graph r u W P P ( 6 / ) (,) i the cotext of graph Here we fid that W is also sallest ordered set such that r( u / W ) r( u / W ); i the cotext of P P, i ; because if we take i W ; the there exist at least two vertices of P havig sae etric represetatio with respect to W i the cotext of graph P P Ad i sae way as case IV (a); W is a sallest ordered set such that (i) r( vi / W) r( v / W) i the cotext of graph P P ; i (ii) r( vi / W ) (,,,,) i i the cotext of graph P P The W WW is a basis for P P ad ( P P ) W W ( P P ) ( P P ) (9) (b) If is odd, the siilarly W { u, u4} ad W { v3, v5, v7, v }; are the sallest ordered sets i P ad P respectively such that (i) r( ui / W) r( u / W) i the cotext of graph P P ; i (ii) r( vi / W ) (,,,,) i i the cotext of graph P P (iii) r( vi / W) r( v / W) i P P i So W W W is a basis for P P ad ( P P ) ( P P ) (0) By (3) ad (4) ( P P) ; 4 6 ad 7 Case VI: If, 7 (a) If ( or ) is eve ad ( or ) is odd Cosider two ordered sets W { u3, u5, u7,, u } adw { v 3, v 5, v 7 v }, the i siilar aer as above we ca show that W W is a basis for P P Therefore

13 Metric Diesio of Soe Graphs uder Joi Operatio 3343 ( ) ( ) P P P P () (b) If both ad are eve, the we ca discuss as above that order sets W { u3, u5, u7,, u } ad W { v3, v5, v7 v } are such that ( P P ) W W ( P P ) ( P P ) () (c) If both ad are odd, the we ca discuss as above that the order sets W { u3, u5, u7,, u } ad W { v3, v5, v7 v } are such that ( P P ) W W ( P P ) ( P P ) (3) (),() ad (3) ( P P) Reark 36: Accordig to Shahida, A T ad M S Suitha [] the etric diesio of oi of two paths Pad P is ( P P) ;, 4 6 Let us take 6 ad 4, the ( P6 P4) 4 6 Now we show that ( P6 P4) 4 Let ui V( P4) ad v V( P6); i 4, 6 Cosider W { u, u, v, v4} V( P6 P4 ); the etric represetatios of V ( P P ) \ W with respect tow are: 6 4 r( u3 / W) (,,,) r( u4 / W) (,,,) r( v / W) (,,,,) r( v3 / W) (,,,) r( v5 / W) (,,,) r( v6 / W) (,,,) Here we fid that all these etric represetatios are differet, therefore ( P P) I theore 35 we have show ( P P ) 3 whe 5ad 3 ad for all other values of ad ( P P) 4 Therefore ( P6P4) 4

14 3344 B S Rawat ad P Pradha Theore 37: Let coected graph, the basis of graph K K be a coplete graph with vertices ad G be ay Gust cotais vertices of Proof Let { v, v, v3, v } be the vertex set of K ad W be a basis for K Suppose basis of K G cotais at ost vertices of K, the we have r( v / W) r( v / W) (,,,); v, v W i i Which is a cotradictio so ay basis of Lea 38: Let K K Gust cotais vertices of G K K be a coplete graph ad G is a coected graph If W is a basis for K G, the W cotais at least vertices Proof Let W be a basis for K G; the basis of K Gust cotai vertices of K ad by lea 3, basis of K G ust cotai at least oe vertex of each graph Therefore W ( K G) Caceres et al [5] have obtaied the etric diesio for the graph P ; {,,3,6} K We have geeralized the result of [5] i the followig theore Theore 39: Let P be a path of vertices ad ; 5 vertices the ( P K) ; 6 8 ; 9 K be a coplete graph with Proof: Let{ u, u, u3,, u } ad { v, v, v3,, v } be the vertex sets of path P ad coplete graph K respectively Followig cases arise: Case I (a): If 5, let us supposew { v, v, v3, v, u, u} V( P K) Now etric represetatio of ay vertex of V ( P K ) \ W with respect to W are respectively:

15 Metric Diesio of Soe Graphs uder Joi Operatio 3345 r( v / W ) (,,,,,) r( u / W ) (,,,,,) 3 r( u / W ) (,,,,, ) 4 Obviously all these represetatios of vertices of graph P K are differet So W is a resolvig set for the graph P K Therefore ( P K) Now if possible, let W { v, v, v3, v, ui}; vig W the at least oe vertex u V ( P) differet fro ui W such that r( v / W) (,,,,) r( u / W) ; a cotradictio So W caot be resolvig set ad therefore it caot be a basis for P K if W ie ( P K ) Therefore ( P K ) Case I (b): If 5; the i siilar aer W { v, v, v, u, u3} ca be proved a basis for the graph P K So ( P K) ; 5 Case II (a): For 6 8 Now if possible W { v, v, v3, v, ui, u }; where W, the at least two vertices u, u i V( P ) havig the sae etric represetatio with respect tow as: k l r( u / W ) (,,,, ) r( u / W ) k or r( u / W ) (,,,,) r( u / W ) k or r( u / W ) (,,,, ) r( u / W ) k or r( u / W ) (,,,,) r( v / W ) k Which are cotradictios; therefore ( P K ) l l l Now let us take W { v, v, v3, v, u3, u5, u6} V( P K); the the etric represetatio of ay vertex of V ( P K ) \ W with respect to W are: r( v / W ) (,,,,,) r( u / W ) (,,,,, ) r( u / W ) (,,,,, ) r( u / W ) (,,,,, ) 4 r( u / W ) (,,,,,) 7

16 3346 B S Rawat ad P Pradha This shows that all vertices of P K have differet etric represetatios with respect to W, so W is a resolvig set ad it is least resolvig set cotaiig 3 i e eleets ( P K ) ; 6 8 Case II (b): If 8; the i siilar aer W { v, v, v, u3, u5, u7} ca be proved basis for P K So ( P K) ; 6 8 Case III (a): If 9 ad is eve Let W { v, v, v, v, u, u, u u } V ( P K ); the the etric represetatio of r( v / W ) (,,,,,,) r( u / W ) (,,,,,, ) r( u / W ) (,,,,,, ) r( u / W ) (,,,,,, ) 4 r( u / W ) (,,,,,, ) 6 r( u / W ) (,,,,,,) 8 r( u / W ) (,,,,,,) vertices of V ( P K ) \ W with respect to W are: These all the represetatios are distict with respect tow So W is a resolvig set for P K ( P K ) W ( P K) Now if we cosider a ordered set W ' such that ( P K ) W ' ad W' W, the we fid that there exist at least two vertices i V( P K ) \ W ' havig the sae etric represetatio with respect to W ' For exaple if W ' W u, the 3 r( u / W ') (,,,,,,,) r( u / W '); a cotradictio If W ' W u, the 5 r( u / W) (,,,,,,,) r( u / W); a cotradictio 5 If W W u ', the 7 r( u / W) (,,,,,,,) r( u / W); a cotradictio 7

17 Metric Diesio of Soe Graphs uder Joi Operatio 3347 So o fially W ' W u, the r( u / W) (,,,,,,,) r( u / W); a cotradictio agai Siilarly for all other cases we get a cotradictio It eas that ay ordered set W ' such that W ' W ca t be a resolvig set for P K It iplies that W is a sallest resolvig set for P K ad ( P K) ; 9 ad is eve () Case III (b): If 9 ad is odd Let W { v, v, v, v, u, u,, u } V( P K ); the as above we ca show that differet vertices of P K have differet etric represetatio with respect to W therefore ( P K ) W ( P K) Now if we take ay ordered set W ' such that W' W, the we observe that there exist at least two vertices i V( P K ) \ W ' havig the sae etric represetatio with respect to W ' So ( P K ) ca t less tha ( P K) Therefore ( P K) ; 9 ad isodd () Now by () ad () iplyig together ( P K) ; 9 ie REFERENCES [] F Harary, RA Melter, O the etric diesio of a graph, Ars Cobiatoria, (976), 9-95 [] G Chartrad, L Eroh, MA Johso, ad OR Oellera, Resolvability i graphs ad the etric diesio of a graph, Discrete Applied Math,05 (000), 99-3

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