Weakly Connected Closed Geodetic Numbers of Graphs

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1 Iteratioal Joural of Mathematical Aalysis Vol 10, 016, o 6, HIKARI Ltd, wwwm-hikaricom Weakly Coected Closed Geodetic Numbers of Graphs Rachel M Pataga 1, Imelda S Aiversario ad Rosalio G Artes, Jr Departmet of Mathematics ad Statistics College of Sciece ad Mathematics Midaao State Uiversity-Iliga Istitute of Techology 900 Iliga City, Philippies Copyright c 015 Rachel M Pataga, Imelda S Aiversario ad Rosalio G Artes, Jr This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited Abstract Give a coected simple graph G ad S V (G), the geodetic closure I G [S] of S is the set of all vertices lyig o some u-v geodesic where u ad v are i S I this paper, select vertices of G sequetially as follows: select a vertex v 1 ad let S 1 = {v 1 } Select a vertex v v 1 ad let S = {v 1, v }, the determie I G [S ] If I G [S ] V (G), the successively select a vertex v i / I G [S i 1 ] ad let S i = {v 1, v,, v i } for i = 1,,, k The determie I G [S i ] A subset S of V (G) is called a weakly coected closed geodetic set of G if the selectio of vertex v k i the give maer yields I G [S k ] = V (G) ad S w is coected, where S w = N[S], E w with E w cosists of edges uv E(G) such that u S or v S The miimum cardiality of weakly coected closed geodetic set is called the weakly coected closed geodetic umber wcg(g) of G I this paper, the weakly coected closed geodetic sets of some commo graphs ad graphs resultig from the joi of graphs are characterized Also, the weakly coected closed geodetic umbers of these graphs are determied 1 This research is fuded through the DOST-ASTHRDP

2 58 Rachel M Pataga, Imelda S Aiversario ad Rosalio G Artes, Jr Keywords: closed geodetic umber, weakly coected closed geodetic set, weakly coected closed geodetic umber 1 Itroductio The cocept o weakly coected closed geodetic umbers of graphs follows from the defiitio of closed geodetic umbers of graphs itroduced by Buckley ad Harary i [] The cocept ivolves closed geodetic closure of a set S V (G) of a graph G deoted by I G [S] The idea evolved from two classes of graphical games called achievemet ad avoidace games The first player A chooses a vertex v 1 of V (G) The secod player B the selects v v 1 ad determies I G [S ] for S = {v 1, v } If I G [S ] V (G), the A picks v 3 / I G [S ] for S 3 = {v 1, v, v 3 } I geeral, A ad B alterately select a ew vertex i this maer The first player who selects a vertex v k / I G [S k 1 ] such that I G [S k ] = V (G) for S k = {v 1, v,, v k }, wis the achievemet game, i the avoidace game he is the loser The ideas o weakly coected geodetic umbers of graphs ad the closed geodetic umbers of graphs have led the researchers to study o weakly coected closed geodetic umbers of graphs The researchers fid it iterestig to study the closed geodetic set S of a coected simple graph G that gives the weakly coected geodetic umbers of graphs where S w is coected Give two vertices u ad v i a coected graph G the distace d G (u, v) betwee u ad v i G is the legth of a shortest path joiig u ad v i G A u-v path of legth d G (u, v) is also referred to as u-v geodesic The eighborhood of v V (G) is the set N G (v) = N(v) = {u V (G) : uv E(G)} If S V (G), the the ope eighborhood of S is the set N G (S) = N(S) = v S N G(v) The closed eighborhood of S is N G [S] = N[S] = S N(S) For every two vertices u ad v of G, the symbol I G [u, v], where u, v V (G) deotes the iterval cotaiig u, v ad all vertices lyig i some u-v geodesic A subset S of V (G) is a geodetic set of G, deoted by g-set, if I G [S] = V (G), where I G [S] = {I G [u, v] : u, v S} I G [S] is called the geodetic closure of S A geodetic set of smallest cardiality is called a geodetic basis of G Heceforth, the set I G (u, v) deotes the set I G [u, v]\ {u, v} The set S is a closed geodetic cover of G if S = {v 1, v,, v k } ad is obtaied by choosig the vertices v 1, v,, v k such that the followig hold: 1 v 1 v ; v i / I G [S i 1 ] for 3 i k; ad 3 I G [S k ] = V (G), where S i = {v 1, v,, v i } for all i = 1,,, k

3 Weakly coected closed geodetic umbers of graphs 59 If S V (G) satisfies (1) ad () above, the S is a closed geodetic subset of V (G) The collectio of all closed geodetic covers of G is deoted by C (G) The closed geodetic umber of G, is give by cg(g) = mi { S : S C (G)} A set S C (G) with S = cg(g) is called the closed geodetic basis of G ad is deoted by cgb(g) Let S V (G) The symbol N[S], E w deotes the weakly iduced subgraph of G with vertex set N[S] ad whose edge set is E w = {uv V (G) : u S or v S} A set S is called a weakly coected closed geodetic set of G, deoted by wcg-set, if it satisfies the followig properties: 1 S C (G) S w is coected where S w = N[S], E w The miimum cardiality of a weakly coected closed geodetic set is called the weakly coected closed geodetic umber of G, deoted by wcg(g) I a weakly coected closed geodetic set S, for every v S, there exists u S such that d G (u, v) Moreover, if {S 1, S,, S k } is the sequece correspodig to the weakly coected geodetic set S, S i w is coected for each i = 1,,, k Cosider the graph i Figure 1 G : u 5 u 1 u u u 3 Figure 1: A graph G of order 5 ad size 6 The set S = {u 1, u, u } is a weakly coected closed geodetic set of graph G Moreover, each -elemet subset S of the vertex set of G has the property that I G [S] is properly cotaied i V (G) Therefore, wcg(g) = 3

4 60 Rachel M Pataga, Imelda S Aiversario ad Rosalio G Artes, Jr Mai Results Throughout this paper, we deote by F G the family of all weakly coected closed geodetic sets i a graph G Clearly, every weakly coected closed geodetic set of a coected graph G is a closed geodetic set A cosequece of this is give as a remark Remark 1 For ay coected graph G, cg(g) wcg(g) Remark For ay coected otrivial graph G of order, cg(g) wcg(g) A vertex v i a graph G is a extreme vertex if the subgraph iduced by its eighborhood is complete The set of extreme vertices is deoted by Ext(G) Sice every extreme vertex is a ed-vertex of every geodesic cotaiig it, we have the followig remark Remark 3 Let G be a graph ad S F G The Ext(G) S It is iterestig to ote that every vertex of a complete graph K is a elemet of Ext(G) as stated by the followig remark Remark Every vertex of a complete graph K is a extreme vertex Aiversario, Jamil ad Caoy i [1] established the followig theorem This theorem shows that for ay coected graph G, cg(g) = if ad oly if G = K This result is used to prove the ext result Theorem 5 [1] Let G be a coected graph with V (G) = The cg(g) = if ad oly if G = K Usig the above results, we ow established the weakly coected closed geodetic umber of a complete graph K Corollary 6 For ay atural umber, wcg(k ) = Proof: By Remarks ad 3, Ext(K ) = V (K ) S V (K ), it follows that S = However, wcg(k ) S = The by Theorem 5 ad Remark, cg(k ) = wcg(k ) Thus combiig the two results we have, wcg(k ) = The followig theorem gives the formula o how to get wcg(c ) for

5 Weakly coected closed geodetic umbers of graphs 61 c 1 c c c 3 c 5 c Figure : A cycle C Theorem 7 Let The wcg(c ) = + 1 Proof : Cosider the cycle C i Figure There are two cases to cosider Case 1: is eve Whe is eve, say = k, for some iteger k, let S 1 = {c 1 } We select those vertices whose distace from its precedig vertices is exactly i order to maitai the miimum cardiality ad let S = {c 1, c 3 } which gives I C [S ] = {c 1, c, c 3 } = V (G) Note that c 5 / I C [S ] Let S 3 = {c 1, c 3, c 5 }, the I C [S 3 ] = {c 1, c, c 3, c, c 5 } Sice cycle C is of order = k, we have C as the uio of two paths [c k, c k 1,, c k ] ad [c k, c 1,, c k ] of the same legth k Without loss of geerality, we ca select those cosecutive vertices i the path [c k, c 1,, c k ] whose subscript i is odd for i = 1,,, k, k Hece, if k is odd the we get, S = {c 1, c 3, c 5,, c k } {c k+ } sice I C [c k+, c 1 ] = {c k+,, c k, c 1 } ad S C (G) Same process is applied, if k is eve we have S = {c 1, c 3, c 5,, c k 1 } {c k+1 } Thus, I C [S] = V (C ) ad S w is coected Therefore, S = {c 1, c 3,, c k } {c k+ } or {c 1, c 3,, c k 1 } {c k+1 } = k + 1 k + 1 or + 1 k = + 1 ( = + 1 = + 1 Now, wcg(c ) + 1 Without loss of geerality, suppose k is eve, we have S = {c 1, c 3, c 5,, c k 1 } {c k+1 } where S = + 1 Suppose we let

6 6 Rachel M Pataga, Imelda S Aiversario ad Rosalio G Artes, Jr S = {c 1, c 3, c 5,, c k 1 } The S = k = Thus, I C [S ] = {c 1, c, c 3, c,, c k 1 } V (C ) Therefore, S / C (C ) O the other had, suppose we let S \ {c i } for ay i = 1, 3, 5,, k 1 The S = k = Sice I C [c k+1, c 1 ] = {c k+1,, c k, c 1 }, hece, I C [S ] = {c 1, c, c 3, c,, c k 1, c k,, c k } = V (C ) Thus, S C (C ) However, for i j, d C (c i, c j ) for ay j = 1, 3, 5, k 1, k +1 But c i / S ad so vertex c i is a isolated vertex i S w This implies that S w is discoected Cosequetly, S / F C Therefore, wcg(c ) = + 1 Case : is odd Whe is odd, say = k + 1, for some iteger k Let S 1 = {c 1 } The we select aother vertex c 3 c 1 for S = {c 1, c 3 } where I C [S ] = {c 1, c, c 3 } Sice cycle C is of order = k + 1, it is the uio of two paths [c k+1, c 1, c,, c k+1 ] ad [c k+1, c k+,, c k, c k+1 ] of legths k + 1 ad k, respectively We select those vertices whose subscripts i s are odd i the path [c k+1, c 1, c,, c k+1 ] Thus, S = {c 1, c 3, c 5,, c k } {c k+ } with I C [c k+, c 1 ] = {c k+,, c k, c k+1, c 1 } or S = {c 1, c 3, c 5,, c k 1 } {c k+1 } with I C [c k+1, c 1 ] = {c k+1,, c k, c k+1, c 1 } whe k is odd or eve, respectively So, I C [S] = V (C ) Hece, S C (C ) for which S w is coected Hece, S = {c 1, c 3,, c k } {c k+ } or {c 1, c 3,, c k 1 } {c k+1 } = k + 1 k + 1 or + 1 k = + 1 ( = + 1 = + 1 Thus, S = + 1 wcg(c ) We are left to show that wcg(c ) + 1 Followig the proof of case 1, we have show that for ay proper subset S of S, S is ot a wcg-set of cycle C Thus, wcg(c ) = S = + 1

7 Weakly coected closed geodetic umbers of graphs 63 Therefore for, wcg(c ) = + 1 Theorem 8 Let The + 1 wcg(p ) = Proof: Let V (P ) = {p 1, p,, p } as show i Figure 3 p 1 p p 3 p 1 p Figure 3: A path P Cosider two cases Case 1: is odd Whe is odd, say = k + 1, for some iteger k Note that Ext(P ) = {p 1, p }, ad by Remark 3, Ext(P ) S Let S 1 = {p 1 } ad S = {p 1, p 3 } where I P [S ] = {p 1, p, p 3 } Cotiuig this process we obtai a set S C (P ) such that S = {p i : i is odd} ad S w is coected Therefore, S = {p 1, p 3,, p k 1, p k+1 } = k + 1 ( ) k = + 1 = k + (k + 1) + 1 = = + 1 Hece, wcg(p ) + 1 Now, let S = S \ Ext(P ) Note that p 1, p Ext(P ) the clearly I P [S ] V (P ) Cosequetly, S / C (P ) O the other had, suppose S = S \ {p i } for i = 3, 5, 7,, k 1 The, I P [S ] = {p 1, p, p 3,, p k+1 } = V (P ) Hece, S C (P ) Observe that for i j, d P (p i, p j ) where j = 1, 3, 5,, k 1, k + 1 This implies that there exists o vertex i S that is adjacet to p i ad sice p i / S, for i = 3, 5, 7,, k 1, the p i is a isolated vertex i S w As a result, S w is discoected Thus, S / F P

8 6 Rachel M Pataga, Imelda S Aiversario ad Rosalio G Artes, Jr Therefore, wcg(p ) = + 1 Case : is eve Whe is eve, say = k, for some iteger k Let S 1 = {p 1 } ad S = {p 1, p 3 } the I P [S ] = {p 1, p, p 3 } = V (P ) So we select aother vertex p 5 / I P [S ] with d P (p 3, p 5 ) = Now, for a path P of order = k, the set S C (P ) ad S w is coected if S = {p i : i is odd} {p =k } Therefore, S = {p 1, p 3,, p k 1 } {p k } = k + 1 ( ) k = + 1 = k + (k + 1) + 1 = ( + 1) + 1 = Hece, S C ( + 1) + 1 (P ) But wcg(p ) S = We are left to show ( + 1) + 1 that wcg(p ) Followig the proof of case 1, we have show that for ay proper subset S of S, S is ot a wcg-set of path P Thus, ( + 1) + 1 wcg(p ) + 1 Therefore, wcg(p ) = We ow show that it is oly possible for a coected graph of order to have a wcg(g) = if ad oly if G is a complete graph Theorem 9 Let G be a coected graph of order The wcg(g) = if ad oly if G = K Proof: Suppose that wcg(g) = Suppose further that G K The there exist x, y V (G) such that d G (x, y) = We costruct a set of vertices i G, S = {v 1, v,, v k } C (G), where k < such that v 1 = x ad v = y Sice I G [v 1, v ] {v 1, v }, we have I G [S] S I fact, I G [S] = V (G) Cosequetly, k < Further, sice G is coected, it follows that S w is also coected Thus, wcg(g) <, a cotradictio to the assumptio Therefore G = K The coverse follows from Corollary 6 The followig theorem is aother cosequece of Remark 3

9 Weakly coected closed geodetic umbers of graphs 65 Theorem 10 For, wcg(k 1, ) = Proof: Write K 1, = K 1 + K ad let A ad B be partite sets of V (K 1, ) where V (K 1 ) = A ad V (K ) = B Sice K 1, is ot a complete graph ad V (K 1, ) = +1 the by Theorem 9, wcg(k 1, ) We oly eed to show that wcg(k 1, ) Let S be a miimum weakly closed coected geodetic set of K 1, Clearly, Ext(K 1, ) = B ad by Remark 3, Ext(K 1, ) S This implies that Ext(K 1, ) = S = wcg(k 1, ) Thus, wcg(k 1, ) Therfore, wcg(k 1, ) = for all The followig result was established by Jamil, Aiversario ad Caoy i [6] This result is used to prove our ext result o the weakly coected closed geodetic sets i the complete bipartite graph K m, Theorem 11 [6] Let G = K m,, where m,, ad let U ad W be the partite sets of G Let S V (G) The S C (G) if ad oly if S is ay of the followig: 1 S = U; S = W ; 3 S = U {w} for some w W ; S = W {u} for some u U As a cosequece of Theorem 11, the followig Lemma is true Lemma 1 Let m, ad let U ad W be the partite sets of K m, A subset S of V (K m, ) is a weakly coected closed geodetic set of K m, if ad oly if S is ay of the followig: 1 S = U; S = W ; 3 S = U {w} for some w W ; S = W {u} for some u U Proof: Suppose S F Km, This implies that S C (K m, ) By Theorem 11, the oly closed geodetic covers of K m, are U, W, U {w} for some w W, ad W {u} for some u U whose weakly iduced subgraphs are coected Coversely, suppose that S is ay of the followig: U, W, U {w} for some w W, ad W {u} for some u U By Theorem 11, S C (K m, ) Sice every vertex i U is adjacet to every vertex i W, the weakly iduced subgraphs they produced are coected Therefore, S F Km,

10 66 Rachel M Pataga, Imelda S Aiversario ad Rosalio G Artes, Jr It is kow that for ay complete bipartite graph K m, for m,, g(k m, ) = mi {m,, } as established by Chartrad, Harary ad Zhag i [] Theorem 13 [] For itegers m,, g(k m, ) = mi {m,, } Usig Lemma 1 ad Theorem 13, the followig theorem ca easily be verified Theorem 1 For itegers m,, wcg(k m, ) = mi {m, } Proof: Write K m, = K m + K ad let U ad W be a partite sets of V (K m, ) By Lemma 1, the oly weakly coected closed geodetic covers of K m, are U, W, U {w} for some w W, ad W {u} for some u U with U = m ad W =, respectively However, oly U ad W are the sets with miimum cardiality Thus, we have wcg(g) = mi { U, W } = mi {m, } The followig corollary characterizes the complete bipartite graph K m, for m,, satisfyig the give coditio The proof is a direct cosequece of Theorems 13 ad 1 Corollary 15 If m,, the wcg(k m, ) = g(k m, ) if ad oly if mi {m, } Proof: Suppose that wcg(k m, ) = g(k m, ), suppose further that mi m, 5 By Theorems 13 ad 1, g(k m, ) = mi {m,, } ad wcg(k m, ) = mi {m, }, respectively Sice mi {m, } 5, it follows that g(k m, ) = while wcg(k m, ) 5 for all m, This implies that wcg(k m, ) g(k m, ), a cotradictio Therefore, mi {m, } Coversely, suppose that mi {m, } Agai, by Theorems 13 ad 1, g(k m, ) = mi {m,, } ad wcg(k m, ) = mi {m, }, respectively Sice mi {m, }, it follows that g(k m, ) while wcg(k m, ) for all m, Therefore, wcg(k m, ) = g(k m, ) It is worth otig that a graph G with order has wcg(g) = 1 if ad oly if the uique graph G is give by G = K 1 + ( m j K j ) as show i the followig result Theorem 16 Let G be a coected graph of order If G = K 1 +( m j K j ), where m j is the umber of copies of K j ad m j, the wcg(g) = 1

11 Weakly coected closed geodetic umbers of graphs 67 Proof: Let G = K 1 + ( m j K j ) with m j The by Theorem 9, 1 wcg(g) Now, let S = V (G)\ {v} where K 1 = {v} The S is a closed geodetic set ad S w = G Thus, wcg(g) S = 1 Therefore, wcg(g) = 1 We preset a theorem that shows the relatioship betwee geodetic set ad weakly coected closed geodetic set Theorem 17 For ay coected graph G of order >, wcg(g) = if ad oly if G has vertices u ad v such that {u, v} is a geodetic set with d G (u, v) = Proof: Let G be a coected graph of order > Suppose that wcg = Let S = {u, v} be a weakly coected closed geodetic set of G The by a property of S, it follows that d G (u, v) However, it is clear to see that d G (u, v) 1 sice G is of order 3 ad I G [S] = V (G) by defiitio of S Hece, d G (u, v) = Therefore, {u, v} is a geodetic set of G with d G (u, v) = Coversely, suppose that G has vertices u ad v such that S = {u, v} is a geodetic set with d G (u, v) = The V ( S w ) = V (G) ad E( S w ) = {ux : x V (G) \ S} {vx : x V (G) \ S} = E w Thus, S is a weakly coected geodetic set of G However, wcg(g) S = By Remark, wcg(g), it follows that wcg(g) = We ow show that for every positive itegers a, b, ad with a b are realizable as the geodetic umber, weakly coected closed geodetic umber ad order of a graph, respectively, of some graph Theorem 18 Give positive itegers a, b, ad with a b, the there exists a coected graph such that V (G) =, g(g) = a, ad wcg(g) = b Proof: Cosider the followig cases: Case 1: = a = b The b = Cosider the graph G i Figure 1 Clearly, S = {x, y} is a miimum g-set of G Also, it is a miimum wcg-set of G Thus, g(g) = wcg(g) = ad V (G) = ( ) + =

12 68 Rachel M Pataga, Imelda S Aiversario ad Rosalio G Artes, Jr z 1 z G : x z 3 y z Figure 1 Case : = a < b = If is eve, the = b 6 If is odd, the = b Cosider the graphs G 1 ad G i Figure v G 1 : x w 1 z 1 w z y G : x w 1 z 1 w z y w b 1 z b 1 Figure w b 1 z b 1 Let S 1 = {x, y} ad S = {x} {z 1, z,, z b 1 } The S 1 is a miimum g-set of G 1 ad G ad S is a miimum wcg-set of G 1 ad G If is eve, the take G = G 1 The V (G) = (b 1) + = b =, g(g) = S 1 = ad wcg(g) = S = b O the other had, if is odd, the take G = G The V (G) = (b 1) + 3 = b + 1 =, g(g) = S 1 = ad wcg(g) = S = b Case 3: = a < b < Let m = b Cosider the graph G give i Figure 3 below Let S 1 = {x, y} ad S = {x} {z 1, z,, z b 1 } The clearly, S 1 is a miimum g-set of G ad S is a miimum wcg-set of G Thus, V (G) = m + (b 1) + = m + b =, g(g) = S 1 = = a ad wcg(g) = S = 1 + (b 1) = b Case : < a = b =

13 Weakly coected closed geodetic umbers of graphs 69 q 1 q q m w 1 z b 1 z 1 w z G : x w b 1 Figure 3 y Cosider the graphs G 1 ad G i Figure x 1 x 1 G 1 : x x x a 1 z y v 1 v v a G : x x x a z y v 1 v v a Figure If is eve, the a = implies that = a Let G = G 1 The V (G) = (a 1)+(a )+3 = a = Sice S = {x, z} {v 1, v,, v a } is both a g-set ad a wcg-set of G, it follows that g(g) = wcg(g) = S = (a ) + = a If is odd, the a = 1 implies that = a + 1 Set G = G The S = {x, z} {v 1, v,, v a } is both a g-set ad a wcg-set of G Therefore, V (G) = a + (a ) + 3 = a + 1 = ad g(g) = wcg(g) = a Case 5: < a = b < Let m = b + 1 Cosider the graph G 1 give i Figure 5 Let S = {x, z} {v 1, v,, v b } The S is both a miimum g-set ad wcg-set of G Thus, V (G) = m + (b ) + 3 = m + b 1 =, g(g) = wcg(g) = S = + (b ) = b = a

14 70 Rachel M Pataga, Imelda S Aiversario ad Rosalio G Artes, Jr q 1 q q m q 1 q q m G 1 : x w 1 w w b y z v 1 v v b w 1 z 1 w z y G : x w b a z b a v 1 v v a 1 Figure 5 Figure 6 Case 6: < a < b Let m = b + a 1 Cosider the graph G i Figure 6 Let S 1 = {x} {v 1, v,, v a 1 } ad S = {x} {z 1, z,, z b a } {v 1, v,, v a 1 } The S 1 ad S is a g-set ad wcg-set of G, respectively Hece, V (G) = m + (a 1) + + (b a) = m + b a + 1 =, g(g) = S 1 = (a 1) + 1 = a ad wcg(g) = S = a + (b a) = b Refereces [1] IS Aiversario, FP Jamil ad SR Caoy Jr, The Closed Geodetic Numbers of Graphs, Utilistas Mathematica, 7 (007), 3-18 [] FT Buckley ad FL Harary, Distace i Graphs, Redwood City, CA, Addiso-Wesley, 1990 [3] FT Buckley ad FL Harary, Geodetic games for graphs, Questioes Mathematicae, 8 (1985), [] GR Chartrad, FL Harary ad P Zhag, Geodetic Sets i Graphs, 39 (00), 1-6 [5] FL Harary, Graph Theory, Addiso-Wesley Readig, MA, 1969 [6] FP Jamil, IS Aiversario, ad SR Caoy, Jr, The Closed Geodetic Numbers of the Coroa ad Compositio of Graphs, Utilistas Mathematica, 8 (010), Received: December 1, 015; Published: February 1, 016