A Study on Total Rebellion Number in Graphs

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1 Joural of Iformatics ad Mathematical Scieces Vol. 9, No. 3, pp , 017 ISSN (olie); X (prit) Published by GN Publicatios Proceedigs of the Coferece Curret Sceario i Pure ad Applied Mathematics December -3, 016 Koguadu Arts ad Sciece College (Autoomous) Coimbatore, Tamil Nadu, Idia A Study o Total ebellio Number i Graphs V. Mohaaselvi* 1, P. Shyamala Ato Mary ad A. Moisha 1 1 Departmet of Mathematics, Nehru Memorial College(Autoomous), Trichy, Idia Departmet of Mathematics, Jayaram college of Egieerig ad Techology, Thuraiyur, Idia Correspodig author: avmoishasri@gmail.com esearch Article Abstract. A set V of a graph G = (V, E) is said to be a rebellio set (rb-set) of G, if N (v) N V / (v), for all v, V / ad has o isolated vertices. The total rebellio umber trb(g) is the miimum cardiality of ay total rebellio set i G. A total rebellio set with cardiality trb(g) is deoted by trb(g)-set. I this paper, we defied the total rebellio umber for simple graphs. Also, we determied its tight bouds for some stadard graph ad characterize this parameters. Keywords. ebellio umber ad Total rebellio umber MSC. 05C38; 05C69 eceived: Jauary, 017 Accepted: March 16, 017 Copyright 017 V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. 1. Itroductio A graph G = (V, E) is a fiite udirected coected simple graph with p vertices ad q edges. All the terms defied here are used i the sese of Harary [1]. The word alliace meas a boud or coectio betwee idividuals, families, states or parties. The uio of idividuals i

2 Hedetiemi ad S.T.Hedetiemi i []. Further it was studied by T.W.Hayes, S.T.Hedetiemi ad A.Heig i [3]. For studyig the co-alliace set the authors already itroduced the rebellio ad idepedet cocept by iterchagig the iequalities i the alliace set i [4] & [5]. A set V of a graph G = (V, E) is said to be a rebellio set (rb- set) of G, if N (v) N V/ (v), v ad V/. The rebellio umber rb(g) is the miimum cardiality of ay rebellio set i G. A total domiatig set D of G is a domiatig set such that the iduced subgraph < > has o isolated vertices. The total domiatio umber γ t (G) of G is the miimum cardiality of a total domiatig set of G [6]. I this paper, we study the total property of the rebellio set ad defied the total rebellio umber for simple graphs. Also, we determied its tight bouds for some stadard graph ad characterize this parameters. 766 A Study o Total ebellio Number i Graphs: V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha a alliace is thought of to be stroger tha the sole idividual. Suppose atios at war, the idividual atios are represeted as vertices ad the relatios betwee them are represeted as edges. Alliace i graphs was first itroduced by Kristiase et al. i []. Further it was studied by Hayes et al. i [3]. For studyig the co-alliace set the authors already itroduced the rebellio ad idepedet cocept by iterchagig the iequalities i the alliace set i [4] ad [5]. A set V of a graph G = (V, E) is said to be a rebellio set (rb-set) of G, if N (v) N V / (v), for all v ad V /. The rebellio umber rb(g) is the miimum cardiality of ay rebellio set i G. A total domiatig set D of G is a domiatig set such that the iduced subgraph has o isolated vertices. The total domiatio umber r t (G) of G is the miimum cardiality of a total domiatig set of G [6]. The Total ebellio Number i Graphs I this paper, we study the total property of the rebellio set ad defied the total rebellio umber for simple graphs. Also, we determied its tight bouds for some stadard graph ad.1 Defiitio characterize this parameters. A rebellio set v of a graph G = (V, E) is said to be a total rebellio set. The Total ebellio Number i Graphs (trb-set) of G, if N (v) N V/ (v), v, V/ ad < > has o isolated vertices. The total rebellio umber trb(g) is the miimum cardiality of ay total rebellio set i G. A total rebellio set with cardiality trb(g) is deoted by trb(g)-set. Defiitio.1. A rebellio set V of a graph G = (V, E) is said to be a total rebellio set (trb-set) of G, if N (v) N V / (v), for all v, V / ad has o isolated vertices. The total rebellio umber trb(g) is the miimum cardiality of ay total rebellio set i G. A total rebellio set with cardiality trb(g) is deoted by trb(g)-set. Example... Example G Figure.1 Figure 1 For the graph G i figure.1, the set = {v, v 3, v 6, v 7, v 11, v 1 } is trb set ad hece trb(g) = 6. For the graph G i Figure 1, the set = {v, v 3, v 6, v 7, v 11, v 1 } is trb-set ad hece trb(g) = 6. Theorem.3. For the cycle C,.3 Theorem /, 0,3(mod 4) For the trb(ccycle ) = C /, + 1, 1,(mod 4), > 5 0, 5. /, 0, 3(mod4) trb(c ) = / + 1, 1, (mod4), > 5 0, 5 Proof. Let G be the cycle graph C with atleast three vertices ad be a total rebellio set i G. Let V (G) = {v 1, v, v 3,..., v } ad E(G) = {e 1, e, e 3,..., e } be the vertex ad edge set of G.

3 A Study o Total ebellio Number i Graphs: V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha 767 Case(i): 0,3(mod 4) I this case, the set = {v 3i+, v 3i+1 /i = 1,,..., 4 } is a total rebellio set of G. Sice V / ad for all v, N(v) is ot a subset of, N (v) N V / (v) ad has 4 trb(g) =. (.1) 4 Let be a total rebellio set of G, such that = trb(g). Sice is a rebellio set gives V / ad for all v, N(v) is ot a subset of, N (v) N V / (v). Thus, must cotai atleast vertices. trb(g) =. (.) From (.1) ad (.) we have, trb(g) =. Case(ii): 1,(mod 4) I this case, the set = {v 3i+, v 3i+1 /i = 1,,..., 4 } is a total rebellio set of G. Sice V / ad for all v, N(v) is ot a subset of, N (v) N V / (v) ad has 4 trb(g) = + 1. (.3) 4 Let be a total rebellio set of G, such that = trb(g). Sice is a rebellio set gives V / ad for all v, N(v) is ot a subset of, N (v) N V / (v). Thus, must cotai atleast + 1 vertices. trb(g) = + 1. (.4) From (.3) ad (.4) we have, trb(g) = + 1. Example.4..4 Example C 9 Figure. For the graph C 9 i figure., the set Figure = {v 1, v, v 4, v 5, v 7, v 8 } is a trb-set ad hece trb(g) = 6. For the graph C 9 i Figure, the set = {v 1, v, v 4, v 5, v 7, v 8 } is a trb-set ad hece trb(g) = 6..5 Theorem Joural For of the Iformatics path P, ad 3, Mathematical Scieces, Vol. 9, No. 3, pp , 017 { /, 0, 3(mod4) trb(p ) = / + 1, 1, (mod4)

4 768 A Study o Total ebellio Number i Graphs: V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha Theorem.5. For the path P, 3, { /, 0,3(mod 4) trb(p ) = / + 1, 1,(mod 4) Proof. Let G be the path graph P with atleast three vertices ad be a total rebellio set i G. Let V (G) = {v 1, v, v 3,..., v } ad E(G) = {e 1, e, e 3,..., e 1 } be the vertex ad edge set of G. Case (i): 0,3(mod 4) I this case, the set = {v 3i+, v 3i+1 /i = 1,,..., 4 } is a total rebellio set of G. Sice V /, for all v, N(v) is ot a subset of, N (v) N V / (v) ad has 4 edges. Now, trb(g) N (v) = N. (.5) V/ 4 (v). Thus, must cotai atleast vertices. Let be a total rebellio set of G, such that = trb(g). Sice is a rebellio set gives V / ad for all v, N(v) is ot a subset of, N (v) N V / (v). Thus, must cotai atleast trb(g) = (6) vertices. From (1) ad () we have, trb(g) = trb(g) =. (.6) Case(ii): 1, (mod4) From (.5) ad (.6) we have, trb(g) =. I this case, the set = {v 3i+, v 3i+1 /i = 1,,... 4 } is a total rebellio Case (ii): set of G, 1,(mod sice 4) V/ ad v, N(v) is ot a subset of, N (v) I this case, N the set = {v 3i+, v 3i+1 /i = 1,,..., V/ (v) ad < > has 4 } is a total rebellio set of G. Sice V / 4 edges, ad for all v, N(v) is ot a subset of, N (v) N V / (v) ad has 4 gives trb(g) = trb(g) + 1. = + 1 (7) (.7) 4 4 Let be Let a total berebellio a total rebellio set of G, set suchofthat G, such = that trb(g). Sice = trb(g). is asice rebellio isset a rebellio V / set ad gives for all such v that, N(v) is ot V/ a subset ad vof,, NN(v) (v) is N ot a V / (v). subset Thus, of must gives such that, N cotai atleast (v) N + 1 vertices. V/ (v). Thus, must cotai atleast + 1 vertices. trb(g) = + 1 (8) trb(g) = + 1. (.8) From (1) ad () we have, trb(g) From (.7) ad (.8) we have, trb(g) = = Example.6..6 Example P 4 Figure.3 Figure 3 For the graph P 4 i figure.3, the set = {v, v 3 } is a trb-set ad hece For thetrb(g) graph P= 4. i Figure 3, the set = {v, v 3 } is a trb-set ad hece trb(g) =..7 Theorem For the complete K,trb(G) =, 3.

5 A Study o Total ebellio Number i Graphs: V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha 769 Theorem.7. For the complete K, trb(g) =, 3. Proof. Let G be the complete graph K with atleast three vertices ad be a total rebellio set i G. Let V (G) = {v 1, v, v 3,..., v } ad E(G) = {e 1, e, e 3,..., e 1 } be the vertex ad edge set of G. The set = {v i /i = 1,,..., } is a total rebellio set of G, Sice V / ad for all, N v, N(v) is (v) N ot a subset V/ (v) ad < > has of, N (v) N V / (v) 4 ad edges, has 4 gives trb(g) =. (.9) trb(g) = (9) Let be a total rebellio set of G, such that = trb(g). Sice is a rebellio set gives such Let be a total rebellio set of G, such that = trb(g), sice is a rebellio that V / set ad for all v, N(v) is ot a subset of, N (v) N V / (v). Thus, must cotai atleast gives such that V/ ad v, N(v) is ot a subset of, N (v) vertices. N V/ (v). Thus, must cotai atleast vertices. trb(g) =. trb(g) = (10) (.10) From (.9) ad (.10), we have trb(g) = From (1) ad () we have, trb(g) =.. Example.8..8 Example K 8 Figure.4 Figure 4 For the graph K 8 i figure.4, the set = {v 1, v, v 4, v 5 } is a trb-set ad hece trb(g) = 4. For the graph K 8 i Figure 4, the set = {v 1, v, v 4, v 5 } is a trb-set ad hece trb(g) = 4. Theorem.9. For the complete bipartite graph K m,.9 Theorem (i) trb(k m, ) = +m, whe m, ad eve For the complete bipartite graph K m, (i)trb(k m, ) = +m, whe m, (ii) trb(kad m, eve ) = 0, (ii) whe trb(k m = m, = ) = odd. 0, whe m = = odd Proof. Let G be the complete bipartite graph K m, with atleast four vertices ad be a total rebellio set i G. Let V (G) = {v 1, v, v 3,..., v m+ } ad E(G) = {e 1, e, e 3,..., e m } be the vertex Proof ad edge set of G. Let G be the complete bipartite graph K m, with atleast four vertices ad be a total rebellio set i G. Let V (G) = {v 1, v, v 3,..., v m+ } ad E(G) = {ejoural 1, e, e 3, of..., Iformatics e m } be the advertex Mathematical ad edge Scieces, set of Vol. G. 9, No. 3, pp , 017 6

6 770 A Study o Total ebellio Number i Graphs: V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha Case(i):m ad are eve I this case, the set = {u 1 } {v i /i = 1,,..., } is a total rebellio set of G, sice V/, v, N(v) is ot a subset of, N (v) N V/ (v) ad < > has 4 edges, Case (i): m ad are eve I this case, the set = {u 1 } {v i /i = 1,,..., } is a total rebellio set of G. Sice V /, for all v, N(v) is ot a subset of, N (v) N V / (v) ad has 4 gives + + m trb(g) = + 1 = + + m (.11) + 1 (because m.) Let be a total rebellio set of G, such that = trb(g). Sice is a rebellio set gives Thus, V /, for all v, N(v) is ot a subset of, N (v) N V / (v). Thus, must cotai atleast p must cotai atleast p + 1 vertices. + 1 vertices. p trb(g) = (because p = + m.) From (.11) ad (.1), we have trb(g) = +m. Case (ii): m ad are odd. rb t (K m, ) = 0 whe m = = odd. the result is obviously. Example.10. trb(g) = = = (11) Let be a total rebellio set of G, such that = trb(g), sice is a rebellio set gives V/, v, N(v) is ot a subset of, N (v) N V/ (v). p trb(g) = From (1) ad () we have, trb(g) = +m. case(ii):m ad are odd rb t (K m, ) = 0 whe m = = odd Therefore the result is obviously..10 Example (1) (.1) ( p = + m) K,10 Figure.5 For the graphk,10 i figure.5, the set = u 1, v 1, v, V 3, V 4, V 5 is a trb-set ad hece trb(g) = 6.11 Theorem For the star graph ad bistar graph trb(g) = 0 Proof Figure 5 For the graph K,10 i Figure 5, the set = {u 1, v 1, v, v 3, v 4, v 5 } is a trb-set ad hece trb(g) = 6 Theorem.11. For the star graph ad bistar graph trb(g) = 0. Proof. Sice every iduced subgraph of rebellio set i star ad bistar are havig isolated vartices ad hece total rebellio set ever exist for those graphs. Hece trb(g) = 0. Sice every iduced subgraph of rebellio set i star ad bistar are havig isolated vartices ad hece total rebellio set ever exist for those graphs. Hece trb(g) = 0 7

7 A Study o Total ebellio Number i Graphs: V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha 771 Example.1..1 Example B 6,6 Figure.6 trb(g) = 0 Figure 6.13 Theorem Theorem.13. For the book graph B For the book graph B = [ = + [ 1 ], + 1 ],.. Proof. Let G be the book graph B with atleast four vertices ad be a total rebellio set i G. LetV (G) = {v 1, v, v 3,..., v + } ad E(G) = {e 1, e, e 3,..., e 3+1 } be the vertex ad edge set of G. The set = {v Proof i /i = 1,,..., }, {u i/i = 1,,...,3 } are a total rebellio set of G. Sice V \, for Letall G be v the, book N(v) graph is ot B with a subset atleastof four, vertices N (v) ad N beva (v) total ad rebellio has edges. Now, set i G. LetV (G) = {v 1, v, v 3,..., v + } ad E(G) = {e 1, e, e 3,..., e 3+1 } be the vertex [ ad ] edge set of G. trb(g) = The set = + 1. (.13) {v i /i = 1,,..., },{u i/i = 1,,..., 3 } are a total rebellio set Let be a total of G, rebellio sice set V of, v G, such, Nv that is ot a= subset trb(g). of, Sice N (v) is N a rebellio V (v) set gives V, for all ad v <, >has N(v), edges, is ot a subset of, N (v) N V (v). Thus, must cotai atleast p vertices (because p = + ). gives p + [ ] [ ] trb(g) = trb(g). = + 1 (13) (.14) Let be a total rebellio set of G, such that = trb(g), sice is a rebellio set gives V, v =,N(v), [ + is 1 ] ot. a subset of, N (v) N V (v). From (.13) ad (.14) we have, trb(g) Example.14. Thus, must cotai atleast p vertices. ( p = + ).14 Example p + trb(g) = From (1) ad () we have, trb(g) = [ + 1 ] [ ] (14) 8 B 5 Figure.7 For the graph B 5 i figure.7, thefigure set = 7 {u 1, v 1, u, v, u 3, v 3 } is a trb-set ad hece trb(g) = 6. For the graph B 5 i Figure 7, the set = {u 1, v 1, u, v, u 3, v 3 } is a trb-set ad hece trb(g) = Theorem Joural For the of Iformatics Fa graph Fad, Mathematical 3, Scieces, Vol. 9, No. 3, pp , 017 { trb(f ) = +1, is eve, is odd. +1

8 77 A Study o Total ebellio Number i Graphs: V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha Theorem.15. For the Fa graph F, 3, { +1, is eve trb(f ) =, is odd. +1 Proof. Let G be the fa graph F with atleast four vertices ad be a total rebellio set i G. Let be a total rebellio set of G, such that = trb(g). Sice is a rebellio Let V (G) set = gives {v 1, v, v 3,..., V v +1, } v ad E(G), N(v) = {e 1 is, e ot, e 3 a,..., subset e of } be, the N vertex ad edge set of G. (v) N V. Case (i): Thus, iseve. must cotai atleast p, vertices. I this case, ( p = the set + 1). = {u} {v i+ /i = 1,,..., 4 } {v i+/i=1,,..., 4 } is a total rebellio set of G, sice V /, for all v, N(v) is ot a subset of, N (v) N V / (v) ad has 4 trb(g) = p (16) trb(g) = (.15) From (1) ad 4 () we 4 have, 4trb(G) = +1 Let be a total rebellio set of G, such that = trb(g). Sice is a rebellio set gives V \, Case(ii): for all v, is N(v) oddis ot a subset of, N (v) N V \. Thus, must cotai atleast p, vertices (because p = + 1). I this case, the set = {u} {v i 1 /i = 1,,... } is a total rebellio set of G, sice p V, + v 1, N(v) is ot a subset of, N (v) N V (v) trb(g) ad< = > has. (.16) 4 edges, From (.15) givesad (.16), we have trb(g) = +1. Case (ii): is odd trb(g) = = (17) I this case, the set = {u} {v i 1 /i = 1,,..., } is a total rebellio set of G, Sice V \, for all v Let, N(v) be a is total ot rebellio a subset of set, of N G, (v) such that N V \ (v) = trb(g). ad has 4 trb(g) = = Sice is a rebellio set gives such that,. V ad v, N(v) is ot a (.17) subset of, N (v) N V (v). Thus, must cotai atleast +1 vertices. Let be ( a p total = + rebellio 1) set of G, such that = trb(g). Sice is a rebellio set gives such that, V \ ad for all v, N(v) is ot a subset of, N (v) N V \ (v). Thus, must cotai atleast +1 vertices (because p = + 1.) trb(g) = + 1 (18) + 1 trb(g) =. (.18) From (3) ad (4) we have,trb(g) = +1. From (.17) ad (.18), we have trb(g) = +1. Example Example F 8 Figure.8 Figure 8 For the graph F 8 i figure.8, the set = {u, v 1, v 3, v 6, vs 8 } is a trb-set ad hece trb(g)=5. 10

9 A Study o Total ebellio Number i Graphs: V. Mohaaselvi, P. Shyamala Ato Mary ad A. Moisha 773 For the graph F 8 i Figure 8, the set = {u, v 1, v 3, v 6, v 8 } is a trb-set ad hece trb(g) = 5. Theorem.17. Let G be a (p, q) graph without isolated vertices. The trb(g) = p m if ad oly if G = mk 3. where m is the umber of compoets greater tha or equal to oe. Proof. Let m be the umber of compoets i G. For the ecessary part assume that trb(g) = p m. Whe, m = 1 the result is obvious. Suppose that G mk 3, m > 1, the for the existece of trb-set for each compoet there exist a vertex v such that it is adjacet to at least two vertices say u ad w. The vertex set V U is a total rebellio set which cotradict the value of trb(g) ad hece the result. The sufficiecy is obvious. Theorem.18. A rb(g)-set is a total rebellio set of G if ad oly if for every vertex i G there exist a adjacet vertex i. Proof. Let be a total rebellio set of G. Suppose that there exist a vertex v V (G) such that v is ot adjacet to ay vertex i. If v the i, v is a isolated vertex, which cotradict is a total rebellio set else is ot a rb(g)-set of G, which is also a cotradictio ad hece the result. For the coverse, suppose that the rb(g)-set is ot a total rebellio set, the there exist a isolated vertex w i such that wis ot adjacet to ay vertex i. Which cotradict every vertex i G there exist a adjacet vertex i ad hece the result. Competig Iterests The author declares that he has o competig iterests. Authors Cotributios The author wrote, read ad approved the fial mauscript. efereces [1] F. Harary, Graph Theory, Addiso Wesley, eadig (1969). [] P. Kristiase, S.K. Hedetiemi ad S.T. Hedetiemi, Alliaces i graphs, J. Cobi. Ath. Cobi. Copute. 48 (004), [3] T.W. Hayes, S.T. Hedetiemi ad A. Heig, Global defesive alliace i graphs, Eletor. J. Cobi. 10, esearch paper 47, 003. [4] V. Mohaa Selvi ad P. Shyamala Ato Mary, The rebellio umber i graphs, Iteratioal Joural of Scietific & Egieerig esearch 7(5) (May 016), 56 6 (ISSN ). [5] V. Mohaa Selvi ad P. Shyamala Ato Mary, Natioal coferece o bridgig iovative treds i pure ad applied mathematics, i: Idepedet ebellio Number i Graphs, Baari Amma Istitute of Techology, Sathyamagalam (016). [6]. Kulli ad B. Jaakiram, The total domiatio umber of a graph, Idia J. Pure Appl. Math. 7(6) (Jue 1996),