The Target Set Selection Problem on Cycle Permutation Graphs, Generalized Petersen Graphs and Torus Cordalis

The T arget Set Sele ction Problem on Cycle P erm utation Graphs, Gene ralized P etersen Graphs and T orus Cor dalis Ch un-Ying Chiang ∗ , Liang-Hao Hu ang † , W ei-Ting Hu ang, Hong-Gw a Y eh ‡§ Dep artment of Mathematics, National Centr al Un iversity, T aiwan Abstract In this pap er w e consider a fund amen tal problem in th e area of viral mar- k eting, called T ARGET S ET S ELECTION p roblem. In a a vir al marke tin g s etting, so cial net wo r ks are mo deled b y graphs with p otent ial customers of a new pro d- uct a s v ertices and friend relationships as edges, wh ere eac h vertex v is assigned a threshold v alue θ ( v ). The thresholds r epresen t the differen t laten t tendencies of cus tomers (v ertices) to buy the new pro du ct when their friend (neigh b ors) do. Consider a r ep etitiv e pr o cess on so cial net wo r k ( G, θ ) w h ere eac h vertex v is asso ciated with tw o states, activ e and inactiv e, which indicate whether v is p ersuaded into buyin g the n ew pr o duct. Supp ose w e are given a target set S ⊆ V ( G ). Initia lly , all v ertices in G are in activ e. A t time step 0, w e c ho ose all v ertices in S to b ecome activ e. Then, at ev ery time step t > 0, all ve r tices that w ere activ e in time step t − 1 remain activ e, and we activ ate any v ertex v if at least θ ( v ) of its neigh b ors were activ e at time s tep t − 1. Th e activ ation pro cess terminates when n o m ore ve r tices can get activ ated. W e are int erested in the follo w ing optimization problem, called T ARGET S ET S ELECTION : Finding a target set S of smallest p ossible size that activ ates all ve rtices of G . There is an imp ortant and w ell-studied thr eshold called strict ma jority threshold, w here for ev ery v ertex v in G we ha ve θ ( v ) = ⌈ ( d ( v ) + 1) / 2 ⌉ and d ( v ) is the degree of v in G . In this p ap er, w e consid er the T ARGET S ET S ELECTION problem under strict ma jorit y thresholds and fo cus on three p opular r egular net work structures: cycle p er mutation graphs, generalized P etersen graphs and torus cordalis. Key wor ds: So cial net wo r ks, viral mark eting, infl uence sp reading, ma j ority v oting, dynamos, target set selection, ir rev ersib le k-threshold, m a jority thresh- old, cycle p erm u tation graph, generalized P etersen graph, torus cordalis, tori. ∗ Partially supp or ted by Natio nal Science Council under gr a nt NSC97-26 28-M-0 08-018- MY3. † Partially supp or ted by Natio nal Science Council under gr a nt NSC98-28 11-M-0 08-072. ‡ Partially supp or ted by Natio nal Science Council under gr a nt NSC100-2 811-M- 008-052 § Corresp o nding a utho r (hgyeh@math.ncu.edu.t w) 1 1 In tro ducti on and preliminary results A gr aph G consists of a set V ( G ) of ve rtic es to g ether with a set E ( G ) of unordered pairs of v ertices called e dges . W e often use uv for an edge { u , v } . Tw o v ertices u and v are adjac ent t o eac h other if uv ∈ E ( G ). In a viral mark eting setting, a so cial netw o rk ( G, θ ) is a connected g r aph G equipped with thresholds θ : V ( G ) → Z , where eac h ve rtex represen ts a p oten tial customer of a new pro duct, a nd eac h edge indicates that the t wo p eople are friends. The thresholds represen t the differen t latent tendencies of v ertices (customers) to buy the new pro duct when their neighbors (f riends) do. There a re three ty p es of imp ortant and w ell-studied thresholds called k -c onstant thr eshold , majority thr eshold and s trict majori ty thr e s h old . In a k -constan t threshold, we hav e θ ( v ) = k for all v ertices v of G , and ( G, θ ) is abbreviated to ( G, k ). In a ma jority threshold for ev ery v ertex v in G we ha v e θ ( v ) = ⌈ d ( v ) / 2 ⌉ , while in a strict ma jority threshold we ha v e θ ( v ) = ⌈ ( d ( v ) + 1) / 2 ⌉ , where d ( v ) is t he degree of v in G . In a so cial net w o rk ( G, θ ), ev ery v ertex in G ha s its own color whic h is ei- ther black or white, where black v ertices represen t active vertice s, and white vertice s represen t inactive vertice s. Giv en a set S ⊆ V ( G ), consider the follo wing rep etitiv e pro cess on ( G, θ ) called activation pr o c ess in ( G , θ ) starting at tar get set S . Initially (at time 0), set all v ertices in S to b e blac k (with all o ther v ertices white). After that, at eac h time step, the states of v ertices are up dat ed according to the follow ing rule: P arallel up dating rule : All inactiv e v ertices v that ha ve a t least θ ( v ) already- activ e neigh b ors b ecome active . The a ctiv atio n pro cess terminates when no more ve rt ices can get activ ated. The set of v ertices that are active at the end of the pro cess is denoted b y [ S ] G θ . If F ⊆ [ S ] G θ , then we say that the t a rget set S in fluenc es F in ( G , θ ). W e are intereste d in the following optimization problem, called T ARGET S ET S ELECTION : Finding a t a rget set S of smallest p ossible size that influences a ll v ertices in the so cial netw or k ( G, θ ), that is [ S ] G θ = V ( G ) (such set S is called a mi n imum se e d or an optimal tar get set f or ( G, θ ) ) . W e define min-seed( G, θ ) = min {| S | : S ⊆ V ( G ) and [ S ] G θ = V ( G ) } . The T AR GET S ET S ELECTION problem and some of its v arian ts w ere introduced and studied in [6, 9, 10, 16, 17, 20, 21, 22, 23, 2 4]. It is not surprising that T ARGET S ET S ELECTION is NP-complete in general. Pele g [21] prov ed that it is NP-hard to compute the optimal target set for ma jority thresholds . In k - constan t threshold setting, D r eyer and Rob erts [10] sho w ed that it is NP-har d to compute the min-seed( G, k ) for any k ≥ 3, and Chen [6] sho w ed that it is also NP-hard to compute min-seed( G, 2). More surprising is the fact that min-seed( G, θ ) is extremely hard to appro ximate. F or any 2 graph G with ma jority t hr esholds θ , Chen [6] prov ed that min-seed( G, θ ) cannot b e approximated within the ratio O (2 log 1 − ǫ n ) for an y fixed constant ǫ > 0, unless N P ⊆ D T I M E ( n poly l og ( n ) ), where n = | V ( G ) | . V ery little is kno wn ab out the exact v alue of min-seed( G, θ ). Related results can b e found in [1, 2 , 3, 6, 7, 1 0, 12, 13, 14, 15, 18, 19, 20, 21], where min- seed( G, θ ) has b een inv estigated under different threshold mo dels for differen t ty p es of net work structure G : b ounded treewidth graphs, trees, blo c k-cactus graphs, chordal graphs, Hamming g r aphs, c hordal ring s, tori, meshes, butterflies. In the curren t pap er, we consider T ARGET S ET S ELECTION pro blem under strict ma jority thresholds and fo cus on three p opular netw ork structures: cycle p ermutation graphs, generalized Pe tersen graphs and torus corda lis. Consider tw o identic a l disjoin t copies G 1 and G 2 of a graph G with p ve rt ices ( p ≥ 4), suc h that V ( G 1 ) = { v 1 , v 2 , . . . , v p } and V ( G 2 ) = { u 1 , u 2 , . . . , u p } , where v i and u i are cor r esp o nding v ertices for eac h i . F or a p ermutation π on { 1 , 2 , . . . , p } , the π -p ermutation gr aph of G , denoted b y P π ( G ), consists of G 1 and G 2 along with p additional edges v i u π ( i ) , i = 1 , 2 , . . . , p . Note that the graph P π ( G ) dep ends not only on the c hoice o f the p erm utation π but on the particular lab eling o f G as w ell. P erm utat io n graphs were in tro duced in [4, 5]. The n -p ath P n is the g raph ha ving V ( P n ) = { x 1 , x 2 , . . . , x n } and E ( P n ) = { x 1 x 2 , x 2 x 3 , . . . , x n − 1 x n } . The n -cycle C n is the gr a ph having V ( C n ) = V ( P n ) and E ( C n ) = E ( P n ) ∪ { x n x 1 } . If G is a cycle, then P π ( G ) is also called a cycle p erm utation gr aph . As examples, cycle p erm utation graphs P π ( C 5 ) are depicted in Figure 1. Figure 1. There are four cycle p erm utatio n graphs for P π ( C 5 ). F or m ≥ 3 and 1 ≤ s ≤ ⌊ m − 1 2 ⌋ , the gener alize d Peterse n gr aph P ( m, s ) is defined to b e the graph with v ertex set V ( P ( m, s )) = { v 1 , v 2 , . . . , v m , u 1 , u 2 , . . . , u m } and edge set E ( P ( m, s )) = { v i v i +1 , u i v i , u i u i + s : i = 1 , 2 , . . . , m } , 3 where the subscripts ar e r ead mo dulo m . These gr a phs w ere in tro duced b y Co xeter [8] and named b y W atkins [26]. As examples, generalized Peters en graphs P (5 , 2), P (10 , 2), and P (10 , 4) are depicted in Figure 2. The connection b etw een generalized P etersen graphs and cycle p erm utatio n graphs is give n in [25]. By the results in [25], w e see that P (10 , 4) is not a cycle p erm utation graph. Clearly , there are tw o cycle p erm utatio n graphs in Figure 1 which are not generalized P etersen graphs. Figure 2. P (5 , 2) ∼ = P etersen gra ph (left), P (10 , 2) ∼ = do decahedral graph (middle), and P (10 , 4) (rig h t). The m × n tor oidal mesh C m  C n is the graph with vertex set { ( i, j ) : 1 ≤ i ≤ m, 1 ≤ j ≤ n } , where t he neighbor s o f ( i, j ) are ( i − 1 , j ) , ( i + 1 , j ) , ( i, j − 1) , ( i, j + 1). Here the a rithmetic in the first co o rdinate is mo dulo m and in the second co o rdinate mo dulo n . The m × n torus c or dalis C m ⊘ C n and m × n toroidal mesh C m  C n ha ve the same v ertex set. The edge set of C m ⊘ C n is almost the same as C m  C n , except that the edge ( i, n )( i, 1) is replaced by t he edge ( i, n )( i + 1 , 1) for 1 ≤ i ≤ m . The m × n torus serp entinus C m ⊗ C n is almost the same as C m ⊘ C n , except that the edge (1 , j ) ( m, j ) is replaced by the edge (1 , j )( m, j + 1) for 1 ≤ j ≤ n . As examples, C 4  C 3 , C 4 ⊘ C 3 and C 4 ⊗ C 3 are depicted in F igure 3. Figure 3. C 4  C 3 (left), C 4 ⊘ C 3 (middle), and C 4 ⊗ C 3 (righ t). In or der to study the minimum seeds for ( G, θ ) w e in tro duce a sequen tial vers ion of activ ation pro cess in ( G, θ ), called se quential activation pr o c ess , whic h emplo ys the follo wing rule instead of the parallel up dating rule: 4 Sequen tial up dating rule: Exactly one of inactiv e v ertices that ha ve at least θ ( v ) already-activ e neigh b or s b ecomes a ctiv e. The pro of of the follo wing lemma is straigh tf o rw ard and so is omitted. In the sequel, Lemma 1 will b e used without explicit reference to it. Lemma 1 A minimum se e d for ( G, θ ) under se quential up dating rule is als o a min- imum se e d fo r ( G, θ ) under p ar al lel up dating rule, and vic e versa. Consider a sequen tial a ctiv atio n pro cess on ( G, θ ) starting from a target set S . In this pro cess, if v 1 , v 2 , . . . , v r is the order that v ertices in [ S ] G θ \ S b ecome blac k, then [ v 1 , v 2 , . . . , v r ] is called the c onvinc e d se quenc e of S on ( G, θ ). In order to describ e a con vinced sequence , w e in tro duce an op erat ion ⊔ on convinc ed subsequences. Let α = [ v 1 , v 2 , . . . , v r ] and β = [ u 1 , u 2 , . . . , u s ]. Then α ⊔ β is defined as α ⊔ β = [ v 1 , v 2 , . . . , v r , u 1 , u 2 , . . . , u s ] , and for a list of con vinced subsequence s { α i,j } 1 ≤ i ≤ ℓ, 1 ≤ j ≤ k , the sequences ⊔ k i =1 α i,j and ⊔ ℓ j =1 ⊔ k i =1 α i,j are defined to b e ⊔ k i =1 α i,j = α 1 ,j ⊔ α 2 ,j ⊔ · · · ⊔ α k ,j and ⊔ ℓ j =1 ⊔ k i =1 α i,j = ⊔ ℓ j =1 ( ⊔ k i =1 α i,j ) . In Section 2, w e precisely determine an opt ima l t arget set for a so cial netw o r k ( G, θ ) when G is a cycle p ermutation graph, and when G is a generalized P etersen graph. In [14], Flo cchini et al. constructed the fo llo wing b ounds on the size of a min- im um seed for toro idal mesh C m  C n , torus cordalis C m ⊘ C n , and torus serpentin us C m ⊗ C n under strict ma jorit y thresholds. Theorem 2 ([14]) (a) ⌈ mn +1 3 ⌉ ≤ min-seed( C m ⊘ C n , 3) ≤ ⌈ m 3 ⌉ ( n + 1) . (b) If G is a C m  C n or a C m ⊗ C n , then ⌈ mn +1 3 ⌉ ≤ min- seed( G, 3) ≤ min {⌈ m 3 ⌉ ( n + 1) , ⌈ n 3 ⌉ ( m + 1) } . In Section 3 of this pap er, w e presen t some impro ved upp er b ounds and exact v alues for the para meter min-seed( C m ⊘ C n , 3). These results are summarized in T able 1. The or e ms m ≥ 1 0 n ≥ 6 Φ Theorem 9(a) o dd n ≡ 0 (mo d 3) Φ = mn 3 + 1 Theorem 9(b) ev en n ≡ 0 (mo d 6) Φ = mn 3 + 1 Theorem 9(c) ev en n ≡ 3 (mo d 6) Φ ∈ { mn 3 + 1 , mn 3 + 2 } Theorem 10 n ≡ 1 (mo d 3) Φ ≤ mn 3 + m 6 + 1 Theorem 11 n ≡ 2 (mo d 3) Φ ≤ mn 3 + m 12 + 3 2 Theorem 12 m ≡ 0 (mo d 3) Φ = mn 3 + 1 T a ble 1. New b ounds and exact v alues for min-seed( C m ⊘ C n , 3), where Φ denotes the parameter min-seed( C m ⊘ C n , 3). 5 2 Cycle p erm utatio n graphs and gen eralized P e- tersen graphs F or X ⊆ V ( G ), let G [ X ] denote the induced subgraph of G whose vertex set is X and whose edge set consists of all edges of G whic h ha ve b oth ends in X . The n umber 1 | V ( G ) | P v ∈ V ( G ) d ( v ) is defined to b e the aver age de gr e e of G and is denoted by d ( G ). W e r emark that all low er b ounds in Theorem 2 follo ws immediately from Lemma 4. Theorem 3 ([11]) Every gr aph with aver age de gr e e at le ast 2 k , wh er e k is a p ositive inte ger, ha s an induc e d sub gr aph with minimum de gr e e at le ast k + 1 . Lemma 4 L et G b e a gr aph with n vertic es, m e dges and maximum de gr e e ∆ . If a tar ge t set S ⊆ V ( G ) in fluenc es al l vertic es in the so cial network ( G, k ) , then | S | ≥ m − (∆ − k ) n +1 k . Pro of. Let ¯ S = V ( G ) \ S a nd G \ S = G [ ¯ S ]. Since S influences all v ertices in the so cial net work ( G, k ), the gr a ph G \ S has no induced subgraph with minimum degree at least ∆ − k + 1. By Theorem 3 , it follows that 2(∆ − k ) > d ( G \ S ) = 2 | E ( G \ S ) | | V ( G \ S ) | ≥ 2( m − ∆ | S | ) n − | S | , where the last inequality follows fro m the fact that if e is an edge in E ( G ) but not in E ( G \ S ), then e has an end in S . W e conclude that (∆ − k )( n − | S | ) ≥ m − ∆ | S | + 1 whic h completes the pro o f of the theorem. Theorem 5 F or any p ermutation π and n ≥ 4 , min-seed( P π ( C n ) , 2) = ⌈ n +1 2 ⌉ . Pro of. Let G = P π ( C n ). Supp ose that G consists of tw o disjoin t copies G 1 and G 2 of C n , suc h tha t V ( G 1 ) = { v 1 , v 2 , . . . , v n } , V ( G 2 ) = { u 1 , u 2 , . . . , u n } , E ( G 1 ) = { v 1 v 2 , v 2 v 3 , . . . , v n − 1 v n , v n v 1 } , E ( G 2 ) = { u 1 u 2 , u 2 u 3 , . . . , u n − 1 u n , u n u 1 } a nd E ( G ) = E ( G 1 ) ∪ E ( G 2 ) ∪ { v i u π ( i ) : i = 1 , 2 , . . . , n } . Without loss of generalit y , we mig ht assume t ha t u 1 v 1 ∈ E ( G ). Let H = G [ { v 3 , v 4 , . . . , v n } ]. In the pro of of Prop osition 2 of [10], it is shown that there is a minim um seed S ′ for the so cial net w ork ( H , 2) suc h that | S ′ | = ⌈ ( n − 1) / 2 ⌉ and { v 3 , v n } ⊆ S ′ . Let [ v i 1 , v i 2 , . . . , v i a ] b e the convinc ed sequence of S ′ on ( H , 2), where a = n − 2 − | S ′ | . Cho ose the target set S = S ′ ∪ { u 1 } for ( G, 2). It can easily b e c heck that S can influence all v ertices of V ( G ) \ S in the social net work ( G, 2) by using [ v i 1 , v i 2 , . . . , v i a ] ⊔ [ v 1 , v 2 ] ⊔ [ u 2 , u 3 , . . . , u n ] as the con vinced sequence. It follows that min-seed( G, 2) ≤ 6 | S | = ⌈ ( n + 1) / 2 ⌉ . Moreov er, f rom Lemma 4, it is easy to see tha t min-seed( G, 2) ≥ ⌈ ( n + 1) / 2 ⌉ . This completes the pro of of the theorem. Clearly , if gcd( m, s ) = 1 , then the generalized P etersen graph P ( m, s ) is a cycle p erm utatio n graph, a nd w e see at o nce that the following corollary holds. In Theorem 7 w e further sho w that Corollar y 6 holds if w e drop the h yp othesis gcd( m, s ) = 1. Corollary 6 If gcd( m, s ) = 1 , then min- seed( P ( m, s ) , 2) = ⌈ m +1 2 ⌉ . Theorem 7 F or m ≥ 3 a n d 1 ≤ s ≤ ⌊ m − 1 2 ⌋ , min-seed( P ( m, s ) , 2) = ⌈ m +1 2 ⌉ . Pro of. Let G = P ( m, s ) . Supp o se that V ( G ) = { v 1 , v 2 , . . . , v m , u 1 , u 2 , . . . , u m } and E ( G ) = { v i v i +1 , u i v i , u i u i + s : i = 1 , 2 , . . . , m } , where the subscripts are read mo dulo m . Let H b e the g raph G [ { v s +1 , v s +2 , . . . , v m − s } ]. Since H is a ( m − 2 s )-path, b y the pro of of Prop osition 2 in [10], w e get a minim um seed S ′ for the so cial net work ( H , 2) suc h that | S ′ | = ⌈ ( m − 2 s + 1) / 2 ⌉ and { v s +1 , v m − s } ⊆ S ′ . Let [ v j 1 , v j 2 , . . . , v j a ] b e the con vinced sequence of S ′ on ( H , 2), where a = m − 2 s − | S ′ | . Consider S = S ′ ∪ { u 1 , u 2 , . . . , u s } as a t arget set for ( G, 2). Since u m − s + i is adjacen t to b o th u i and u m − 2 s + i for i ∈ { 1 , 2 , . . . , s } and u s + j is adjacen t to b oth u j and v s + j for j ∈ { 1 , 2 , . . . , m − 2 s } , w e see that S can influence all v ertices of V ( G ) \ S in t he so cial net w ork ( G, 2) by using [ v j 1 , v j 2 , . . . , v j a ] ⊔ [ v s , v s − 1 , . . . , v 1 ] ⊔ [ u s +1 , u s +2 , . . . , u m − s ] ⊔ [ u m − s +1 , u m − s +2 , . . . , u m ] ⊔ [ v m − s +1 , v m − s +2 , . . . , v m ] as the con- vinced sequence. Therefore min-seed( G, 2 ) ≤ | S | = | S ′ | + s = ⌈ ( m + 1) / 2 ⌉ . Moreo ver, b y Lemma 4, it can b e seen that min-seed( G, 2) ≥ ⌈ ( m + 1) / 2 ⌉ . This complete the pro of of the theorem. 3 T orus cordalis Theorem 8 min-seed( C m ⊘ C 3 , 3) = m + 1 for an y m ≥ 3 . Pro of. L et G = C m ⊘ C 3 . Fir st we show t hat min-seed( G, 3) ≤ m + 1 by giving a target set S ⊆ V ( G ) whic h influences a ll v ertices of G . Denote by S 1 and S 2 the sets { (2 i + 1 , 1) : 0 ≤ i ≤ ⌊ m − 1 2 ⌋} a nd { (2 i, 2) : 1 ≤ i ≤ ⌊ m 2 ⌋} , resp ectiv ely . Se t S = S 1 ∪ S 2 ∪ { (1 , 3) } . Let α 1 = [(2 , 1) , (4 , 1) , . . . , (2 ⌊ m 2 ⌋ , 1)], α 2 = [(1 , 2) , (3 , 2) , . . . , (1 + 2 ⌊ m − 1 2 ⌋ , 2)] and α 3 = [(2 , 3) , (3 , 3) , . . . , ( m, 3 )]. It is straigh t f orw ard to see that α 1 ⊔ α 2 ⊔ α 3 is a con vinced sequence of S on ( G, 3) (see F igure 1 in App endix for a graphical illustration of this conv inced seque nce). By the low er b ound of min-seed ( C m ⊘ C 3 , 3) in Theorem 2(a), w e see that min-seed( C m ⊘ C 3 , 3) = m + 1. 7 Theorem 9 L et s ≥ 2 b e an inte ger. (a) If m ≥ 5 is an o dd inte ger, then min- seed( C m ⊘ C 3 s , 3) = ms + 1 . (b) If m ≥ 8 is an even inte ger and s is an even inte ger, then min-seed( C m ⊘ C 3 s , 3) = ms + 1 . ( c ) If m ≥ 8 is an eve n inte ger and s is an o dd inte ger, then min-seed( C m ⊘ C 3 s , 3) = ms + 1 or ms + 2 . Pro of. Let G = C m ⊘ C 3 s . (a) Denote by S 1 and S 2 the sets ∪ s − 1 j =0 { (1 , 1 + 3 j ) , (2 , 2 + 3 j ) , (3 , 1 + 3 j ) , (4 , 3 + 3 j ) , (5 , 2 + 3 j ) } and ∪ s − 1 j =0 ∪ m − 7 2 i =0 { (6 + 2 i, 3 + 3 j ) , (7 + 2 i, 2 + 3 j ) } , respectiv ely . Let S = S 1 ∪ S 2 ∪ { (4 , 1) } . In the social netw ork ( G, 3), it is straigh tforward to c hec k that the target set S can influence all v ertices in V ( G ) \ S b y using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 , where α 1 = ⊔ s − 1 j =0 [(1 , 2 + 3 j ) , (2 , 1 + 3 j )], α 2 = ⊔ s − 1 j =0 ⊔ m − 7 2 i =0 [(5 + 2 i, 3 + 3 j ) , (6 + 2 i, 2 + 3 j )], α 3 = [(5 , 1) , (6 , 1) , (7 , 1) , . . . , ( m, 1)], α 4 = [(4 , 2) , (3 , 2) , (3 , 3) , (2 , 3) , (1 , 3) , ( m, 3)], and α 5 = ⊔ s − 2 j =0 ([( m, 4 + 3 j ) , ( m − 1 , 4 + 3 j ) , . . . , (4 , 4 + 3 j )] ⊔ [(4 , 5 + 3 j ) , (3 , 5 + 3 j ) , (3 , 6 + 3 j ) , (2 , 6 + 3 j ) , (1 , 6 + 3 j ) , ( m, 6 + 3 j )]) (see F igure 2 in App endix fo r a graphical illustration of this con vinced sequence α ). Therefore min-seed( C m ⊘ C 3 s , 3) ≤ | S | = ms + 1 and hence b y Theorem 2(a), we hav e min-seed( C m ⊘ C 3 s , 3) = ms + 1. (b) Denote b y S 1 , S 2 and S 3 the sets ∪ s − 1 j =0 { (1 , 1 + 3 j ) , (2 , 2 + 3 j ) , (3 , 1+ 3 j ) , (4 , 3 + 3 j ) , (5 , 2 + 3 j ) } , ∪ s − 1 j =0 ∪ m − 10 2 i =0 { (6 + 2 i, 3 + 3 j ) , (7 + 2 i, 2 + 3 j ) } and ∪ s − 1 j =0 { ( m − 2 , 1 + 3 j ) , ( m − 1 , 3 + 3 j ) , ( m, 2 + 3 j ) } , resp ectiv ely . Let S = S 1 ∪ S 2 ∪ S 3 ∪ { (4 , 1) } . It can readily b e c heck ed tha t the target set S can influence a ll ve rt ices in V ( G ) \ S by using the convinc ed sequen ce α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 ⊔ α 7 (see Figure 3 in App endix for a graphical illustration of this con vinced sequence α ), where α 1 = ⊔ s − 1 j =0 [(1 , 2 + 3 j ) , (2 , 1 + 3 j )], α 2 = ⊔ s − 1 j =0 ⊔ m − 10 2 i =0 [(5 + 2 i, 3 + 3 j ) , (6 + 2 i, 2 + 3 j ) ], α 3 = [(5 , 1) , (6 , 1 ) , (7 , 1) , . . . , ( m − 3 , 1)] ⊔ [( m − 3 , 3 s ) ], α 4 = [(4 , 2) , (3 , 2 ) , (3 , 3) , (2 , 3) , (1 , 3) , ( m, 3)], α 5 = ⊔ s − 4 2 k =0 ([( m, 4 + 6 k ) , ( m − 1 , 4 + 6 k ) , ( m − 1 , 5 + 6 k ) , ( m − 2 , 5 + 6 k ) , ( m − 2 , 6 + 6 k ) , ( m − 3 , 6 + 6 k )] ⊔ [( m − 3 , 7 + 6 k ) , ( m − 4 , 7 + 6 k ) , . . . , (4 , 7 + 6 k )] ⊔ [(4 , 8 + 6 k ) , (3 , 8 + 6 k ) , (3 , 9 + 6 k ) , (2 , 9 + 6 k ) , (1 , 9 + 6 k ) , ( m, 9 + 6 k )]), α 6 = [( m, 3 s − 2) , ( m − 1 , 3 s − 2) , ( m − 1 , 3 s − 1) , ( m − 2 , 3 s − 1) , ( m − 2 , 3 s )], and α 7 = ⊔ s − 2 2 k =0 ([( m, 1 + 6 k ) , ( m − 1 , 1 + 6 k ) , ( m − 1 , 2 + 6 k ) , ( m − 2 , 2 + 6 k ) , ( m − 2 , 3 + 6 k ) , ( m − 3 , 3 + 6 k )] ⊔ [( m − 3 , 4 + 6 k ) , ( m − 4 , 4 + 6 k ) , . . . , (4 , 4 + 6 k )] ⊔ [(4 , 5 + 6 k ) , (3 , 5 + 6 k ) , (3 , 6 + 6 k ) , (2 , 6 + 6 k ) , (1 , 6 + 6 k ) , ( m, 6 + 6 k )]). 8 Therefore min-seed( C m ⊘ C 3 s , 3) ≤ | S | = ms + 1 and hence b y Theorem 2(a), w e ha v e min-seed( C m ⊘ C 3 s , 3) = ms + 1. (c) D enote b y S 1 , S 2 and S 3 the sets ∪ s − 1 j =0 { (1 , 1 + 3 j ) , (2 , 2 + 3 j ) , (3 , 1 + 3 j ) , (4 , 3 + 3 j ) , (5 , 2 + 3 j ) } , ∪ s − 1 j =0 ∪ m − 10 2 i =0 { (6 + 2 i, 3 + 3 j ) , (7 + 2 i, 2 + 3 j ) } and ∪ s − 1 j =0 { ( m − 2 , 1 + 3 j ) , ( m − 1 , 3 + 3 j ) , ( m, 2 + 3 j ) } , resp ectiv ely . Let S = S 1 ∪ S 2 ∪ S 3 ∪ { (4 , 1) , ( m − 1 , 1) } . It is straig h tforward to chec k that the target se t S can influence all v ertices in V ( G ) \ S b y using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 ⊔ α 7 (see F igure 4 in App endix for a graphical illustration of this convince d sequence α ), where α 1 = ⊔ s − 1 j =0 [(1 , 2 + 3 j ) , (2 , 1 + 3 j )], α 2 = ⊔ s − 1 j =0 ⊔ m − 10 2 i =0 [(5 + 2 i, 3 + 3 j ) , (6 + 2 i, 2 + 3 j ) ], α 3 = [(5 , 1) , (6 , 1 ) , (7 , 1) , . . . , ( m − 3 , 1)], α 4 = [(4 , 2) , (3 , 2 ) , (3 , 3) , (2 , 3) , (1 , 3 ) , ( m, 3 )], α 5 = [( m, 1) , ( m − 1 , 2) , ( m − 2 , 2) , ( m − 2 , 3) , ( m − 3 , 3)], α 6 = ⊔ s − 3 2 k =0 ([( m − 3 , 4 + 6 k ) , ( m − 4 , 4 + 6 k ) , . . . , (4 , 4 + 6 k )] ⊔ [(4 , 5 + 6 k ) , (3 , 5 + 6 k ) , (3 , 6 + 6 k ) , (2 , 6 + 6 k ) , (1 , 6 + 6 k ) , ( m, 6 + 6 k )] ⊔ [( m, 7 + 6 k ) , ( m − 1 , 7 + 6 k ) , ( m − 1 , 8 + 6 k ) , ( m − 2 , 8 + 6 k ) , ( m − 2 , 9 + 6 k ) , ( m − 3 , 9 + 6 k )]), and α 7 = ⊔ s − 3 2 k =0 ([( m, 4 + 6 k ) , ( m − 1 , 4 + 6 k ) , ( m − 1 , 5 + 6 k ) , ( m − 2 , 5 + 6 k ) , ( m − 2 , 6 + 6 k ) , ( m − 3 , 6 + 6 k )] ⊔ [( m − 3 , 7 + 6 k ) , ( m − 4 , 7 + 6 k ) , . . . , (4 , 7 + 6 k )] ⊔ [(4 , 8 + 6 k ) , (3 , 8 + 6 k ) , (3 , 9 + 6 k ) , (2 , 9 + 6 k ) , (1 , 9 + 6 k ) , ( m, 9 + 6 k )]). Therefore min-seed( C m ⊘ C 3 s , 3) ≤ | S | = ms + 2 and hence b y Theorem 2(a), w e ha v e min-seed( C m ⊘ C 3 s , 3) = ms + 1 or ms + 2. Theorem 10 If m ≥ 8 and n ≡ 1 (mo d 3) , then min-seed( C m ⊘ C n , 3) ≤ mn 3 + m 6 +1 . Pro of. Let G = C m ⊘ C n and n = 3 s + 1. The pro of is divided into three cases, according to the parit y o f the tw o inte gers m and s . Case 1. m ≥ 5 is o dd. Let S 1 , S 2 and S 3 denote the sets ∪ s − 1 j =0 { (1 , 2 + 3 j ) , (2 , 3 + 3 j ) , (3 , 2 + 3 j ) , (4 , 4 + 3 j ) , (5 , 3 + 3 j ) } , { (6 , 1) , (8 , 1) , . . . , ( m − 1 , 1) } and ∪ s − 1 j =0 ∪ m − 7 2 i =0 { (6 + 2 i, 4 + 3 j ) , (7 + 2 i, 3 + 3 j ) } , resp ectiv ely . Let S = { (1 , 1) , (2 , 1) , (4 , 1) } ∪ S 1 ∪ S 2 ∪ S 3 . It is straigh tfo r ward to c hec k that the ta rget set S can influence all v ertices in V ( G ) \ S by using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 , where α 1 = ⊔ s − 1 j =0 [(1 , 3 + 3 j ) , (2 , 2 + 3 j )], α 2 = ⊔ s − 1 j =0 ⊔ m − 7 2 i =0 [(5 + 2 i, 4 + 3 j ) , (6 + 2 i, 3 + 3 j )], α 3 = 9 [(3 , 1) , (5 , 1) , (7 , 1) , . . . , ( m, 1)], and α 4 = ⊔ s − 1 j =0 ([( m, 2 + 3 j ) , ( m − 1 , 2 + 3 j ) , . . . , (4 , 2 + 3 j )] ⊔ [(4 , 3 + 3 j ) , (3 , 3 + 3 j ) , (3 , 4 + 3 j ) , ( 2 , 4 + 3 j ) , (1 , 4 + 3 j ) , ( m, 4 + 3 j )]) (see Figure 5 in App endix for a graphical illustration of this convinc ed sequence α ). Therefore min-seed( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 6 + 1 2 . Case 2. m ≥ 8 is eve n a nd s is o dd. Denote b y S 1 , S 2 , S 3 and S 4 the sets ∪ s − 1 j =0 { (1 , 2 + 3 j ) , (2 , 3 + 3 j ) , (3 , 2 + 3 j ) , (4 , 4 + 3 j ) , (5 , 3 + 3 j ) } , { ( 6 , 1) , (8 , 1) , . . . , ( m − 4 , 1) } , ∪ s − 1 j =0 ∪ m − 10 2 i =0 { (6 + 2 i, 4 + 3 j ) , (7 + 2 i, 3 + 3 j ) } and ∪ s − 1 j =0 { ( m − 2 , 2 + 3 j ) , ( m − 1 , 4 + 3 j ) , ( m, 3+ 3 j ) } , respectiv ely . Let S = S 1 ∪ S 2 ∪ S 3 ∪ S 4 ∪{ (1 , 1) , (2 , 1 ) , (4 , 1) , ( m − 1 , 1) } . It can readily b e c hec k ed that the target set S can influence all v ertices in V ( G ) \ S b y using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 ⊔ α 7 (see F igure 6 in App endix for a graphical illustration of this convince d sequence α ), where α 1 = ⊔ s − 1 j =0 [(1 , 3 + 3 j ) , (2 , 2 + 3 j )], α 2 = ⊔ s − 1 j =0 ⊔ m − 10 2 i =0 [(5 + 2 i, 4 + 3 j ) , (6 + 2 i, 3 + 3 j ) ], α 3 = [(3 , 1) , (5 , 1 ) , (7 , 1) , . . . , ( m − 5 , 1)], α 4 = [( m, 1) , ( m, 2) , ( m − 1 , 2) , ( m − 1 , 3) , ( m − 2 , 3) , ( m − 2 , 4) , ( m − 3 , 4)], α 5 = ⊔ s − 3 2 k =0 ([( m − 3 , 5 + 6 k ) , ( m − 4 , 5 + 6 k ) , . . . , (4 , 5 + 6 k )] ⊔ [(4 , 6 + 6 k ) , (3 , 6 + 6 k ) , (3 , 7 + 6 k ) , (2 , 7 + 6 k ) , (1 , 7 + 6 k ) , ( m, 7 + 6 k )] ⊔ [( m, 8 + 6 k ) , ( m − 1 , 8 + 6 k ) , ( m − 1 , 9 + 6 k ) , ( m − 2 , 9 + 6 k ) , ( m − 2 , 1 0 + 6 k ) , ( m − 3 , 10 + 6 k )]), α 6 = [( m − 2 , 1) , ( m − 3 , 1)] ⊔ [( m − 3 , 2) , ( m − 4 , 2) , . . . , (4 , 2)] ⊔ [(4 , 3) , (3 , 3 ) , (3 , 4) , (2 , 4) , (1 , 4) , ( m, 4)], and α 7 = ⊔ s − 3 2 k =0 ([( m, 5 + 6 k ) , ( m − 1 , 5 + 6 k ) , ( m − 1 , 6 + 6 k ) , ( m − 2 , 6 + 6 k ) , ( m − 2 , 7 + 6 k ) , ( m − 3 , 7 + 6 k )] ⊔ [( m − 3 , 8 + 6 k ) , ( m − 4 , 8 + 6 k ) , . . . , (4 , 8 + 6 k )] ⊔ [(4 , 9 + 6 k ) , (3 , 9 + 6 k ) , (3 , 10 + 6 k ) , (2 , 10 + 6 k ) , (1 , 10 + 6 k ) , ( m, 10 + 6 k )]). Therefore min-seed( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 6 . Case 3. m ≥ 8 and s are b oth ev en. Denote b y S 1 , S 2 , S 3 and S 4 the sets ∪ s − 1 j =0 { (1 , 2 + 3 j ) , (2 , 3 + 3 j ) , (3 , 2 + 3 j ) , (4 , 4 + 3 j ) , (5 , 3 + 3 j ) } , { ( 6 , 1) , (8 , 1) , . . . , ( m − 4 , 1) } , ∪ s − 1 j =0 ∪ m − 10 2 i =0 { (6 + 2 i, 4 + 3 j ) , (7 + 2 i, 3 + 3 j ) } and ∪ s − 1 j =0 { ( m − 2 , 2 + 3 j ) , ( m − 1 , 4 + 3 j ) , ( m, 3 + 3 j ) } , r esp ective ly . Let S = S 1 ∪ S 2 ∪ S 3 ∪ S 4 ∪ { (1 , 1) , (2 , 1 ) , (4 , 1) , ( m − 2 , 1) , ( m − 1 , 1) } . It is straigh tforward to chec k that the target set S can influence all ve rtices in V ( G ) \ S b y using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 (see F igure 7 in App endix fo r a graphical illustration of this con vinced sequence α ), where 10 α 1 = ⊔ s − 1 j =0 [(1 , 3 + 3 j ) , (2 , 2 + 3 j )], α 2 = ⊔ s − 1 j =0 ⊔ m − 10 2 i =0 [(5 + 2 i, 4 + 3 j ) , (6 + 2 i, 3 + 3 j ) ], α 3 = [(3 , 1) , (5 , 1 ) , (7 , 1) , . . . , ( m − 3 , 1)] ⊔ [( m, 1)], α 4 = ⊔ s − 2 2 k =0 ([( m − 3 , 2 + 6 k ) , ( m − 4 , 2 + 6 k ) , . . . , (4 , 2 + 6 k )] ⊔ [(4 , 3 + 6 k ) , (3 , 3 + 6 k ) , (3 , 4 + 6 k ) , (2 , 4 + 6 k ) , (1 , 4 + 6 k ) , ( m, 4 + 6 k )] ⊔ [( m, 5 + 6 k ) , ( m − 1 , 5 + 6 k ) , ( m − 1 , 6 + 6 k ) , ( m − 2 , 6 + 6 k ) , ( m − 2 , 7 + 6 k ) , ( m − 3 , 7 + 6 k )]), and α 5 = ⊔ s − 2 2 k =0 ([( m, 2 + 6 k ) , ( m − 1 , 2 + 6 k ) , ( m − 1 , 3 + 6 k ) , ( m − 2 , 3 + 6 k ) , ( m − 2 , 4 + 6 k ) , ( m − 3 , 4 + 6 k )] ⊔ [( m − 3 , 5 + 6 k ) , ( m − 4 , 5 + 6 k ) , . . . , (4 , 5 + 6 k )] ⊔ [(4 , 6 + 6 k ) , (3 , 6 + 6 k ) , (3 , 7 + 6 k ) , (2 , 7 + 6 k ) , (1 , 7 + 6 k ) , ( m, 7 + 6 k )]). W e conclude that min-seed( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 6 + 1. This completes the pro of of the theorem. Theorem 11 If m ≥ 10 , n ≡ 2 (mo d 3) and n ≥ 5 , then min-seed( C m ⊘ C n , 3) ≤ mn 3 + m 12 + 3 2 . Pro of. Let G = C m ⊘ C n , m = 4 t + r and n = 3 s + 2, where t, r , s a re in t egers with 0 ≤ r ≤ 3. The pr o of is divided in to six cases, according to the v alue of r and the parit y of s . Case 1. r = 0 and s is even . In this case, let S 1 , S 2 , S 3 and S 4 denote the sets ∪ s − 1 j =0 { (1 , 3 + 3 j ) , (2 , 4 + 3 j ) , (3 , 3 + 3 j ) } , ∪ t − 3 i =0 { (4 + 4 i, 1) , (6 + 4 i, 1) , (6 + 4 i, 2) } , ∪ s − 1 j =0 ∪ t − 3 i =0 { (4 + 4 i, 5 + 3 j ) , (5 + 4 i, 3 + 3 j ) , (6 + 4 i, 5 + 3 j ) , (7 + 4 i, 3 + 3 j ) } and ∪ s − 1 j =0 { ( m − 4 , 5 + 3 j ) , ( m − 3 , 4 + 3 j ) , ( m − 2 , 3 + 3 j ) , ( m − 1 , 5 + 3 j ) , ( m, 4 + 3 j ) } , resp ective ly . Let S = { (2 , 1) , (3 , 2) , ( m − 4 , 1) , ( m − 3 , 2) , ( m − 1 , 1) , ( m, 2) } ∪ S 1 ∪ S 2 ∪ S 3 ∪ S 4 . It is straigh tfo r ward to c heck that the target set S can influence a ll v ertices in V ( G ) \ S b y using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 ⊔ α 7 ⊔ α 8 (see Figure 8 in App endix for a graphical illustration of this con vinced sequen ce α ), where α 1 = ⊔ s − 1 j =0 [(1 , 4 + 3 j ) , (2 , 3 + 3 j )], α 2 = ⊔ t − 3 i =0 [(7 + 4 i, 1 ) , (7 + 4 i, 2 ) , (5 + 4 i, 1) , (5 + 4 i, 2) , (4 + 4 i, 2)], α 3 = ⊔ s − 1 j =0 ⊔ t − 3 i =0 [(4 + 4 i, 3 + 3 j ) , (5 + 4 i, 5 + 3 j ) , (6 + 4 i, 3 + 3 j ) , (7 + 4 i, 5 + 3 j )], α 4 = [( m − 4 , 2) , ( m − 3 , 1 )], α 5 = ⊔ s − 2 2 k =0 ([( m − 3 , 3 + 6 k ) , ( m − 4 , 3 + 6 k )] ⊔ [( m − 4 , 4 + 6 k ) , ( m − 5 , 4 + 6 k ) , ( m − 6 , 4 + 6 k ) , . . . , (3 , 4 + 6 k )] ⊔ [(3 , 5 + 6 k ) , (2 , 5 + 6 k ) , (1 , 5 + 6 k ) , ( m, 5 + 6 k )] ⊔ [( m, 6 + 6 k ) , ( m − 1 , 6 + 6 k ) , ( m − 1 , 7 + 6 k ) , ( m − 2 , 7 + 6 k ) , ( m − 2 , 8 + 6 k ) , ( m − 3 , 8 + 6 k )]), 11 α 6 = [( m − 2 , 1) , ( m − 2 , 2 ) , ( m − 1 , 2) , ( m, 1)], α 7 = [(3 , 1) , (2 , 2 ) , (1 , 2) , (1 , 1)], and α 8 = ⊔ s − 2 2 k =0 ([( m, 3 + 6 k ) , ( m − 1 , 3 + 6 k ) , ( m − 1 , 4 + 6 k ) , ( m − 2 , 4 + 6 k ) , ( m − 2 , 5 + 6 k ) , ( m − 3 , 5 + 6 k ) , ( m − 3 , 6 + 6 k ) , ( m − 4 , 6 + 6 k )] ⊔ [( m − 4 , 7 + 6 k ) , ( m − 5 , 7 + 6 k ) , ( m − 6 , 7 + 6 k ) , . . . , (3 , 7 + 6 k )] ⊔ [( 3 , 8+ 6 k ) , (2 , 8+ 6 k ) , (1 , 8+ 6 k ) , ( m, 8 +6 k )]). W e conclude that min-seed( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 12 . Case 2. r = 0 and s is o dd. In this case, let S 1 , S 2 , S 3 and S 4 denote the sets ∪ s − 1 j =0 { (1 , 3 + 3 j ) , (2 , 4 + 3 j ) , (3 , 3 + 3 j ) } , ∪ t − 3 i =0 { (4 + 4 i, 1) , (6 + 4 i, 1) , (6 + 4 i, 2) } , ∪ s − 1 j =0 ∪ t − 3 i =0 { (4 + 4 i, 5 + 3 j ) , (5 + 4 i, 3 + 3 j ) , (6 + 4 i, 5 + 3 j ) , (7 + 4 i, 3 + 3 j ) } and ∪ s − 1 j =0 { ( m − 4 , 5 + 3 j ) , ( m − 3 , 4 + 3 j ) , ( m − 2 , 3 + 3 j ) , ( m − 1 , 5 + 3 j ) , ( m, 4 + 3 j ) } , resp ectiv ely . Let S = { (2 , 1) , (3 , 2) , ( m − 4 , 1) , ( m − 3 , 2) , ( m − 2 , 1) , ( m − 1 , 1) , ( m, 2) } ∪ S 1 ∪ S 2 ∪ S 3 ∪ S 4 . It is straig h tforward to chec k that the target se t S can influence all v ertices in V ( G ) \ S b y using the conv inced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 ⊔ α 7 ⊔ α 8 ⊔ α 9 (see Figure 9 in App endix for a g raphical illustration of this con vinced sequence α ), where α 1 = ⊔ s − 1 j =0 [(1 , 4 + 3 j ) , (2 , 3 + 3 j )], α 2 = ⊔ t − 3 i =0 [(7 + 4 i, 1 ) , (7 + 4 i, 2 ) , (5 + 4 i, 1) , (5 + 4 i, 2) , (4 + 4 i, 2)], α 3 = ⊔ s − 1 j =0 ⊔ t − 3 i =0 [(4 + 4 i, 3 + 3 j ) , (5 + 4 i, 5 + 3 j ) , (6 + 4 i, 3 + 3 j ) , (7 + 4 i, 5 + 3 j )], α 4 = [( m − 4 , 2) , ( m − 3 , 1 ) , ( m − 2 , 2) , ( m − 1 , 2) , ( m, 1)], α 5 = [(3 , 1) , (2 , 2 ) , (1 , 2) , (1 , 1)], α 6 = [( m − 3 , 3) , ( m − 4 , 3 )] ⊔ [( m − 4 , 4) , ( m − 5 , 4) , ( m − 6 , 4) , . . . , (3 , 4)] ⊔ [(3 , 5) , (2 , 5) , (1 , 5) , ( m, 5)], α 7 = [( m, 3) , ( m − 1 , 3) , ( m − 1 , 4) , ( m − 2 , 4) , ( m − 2 , 5) , ( m − 3 , 5)], α 8 = ⊔ s − 3 2 k =0 ([( m − 3 , 6 + 6 k ) , ( m − 4 , 6 + 6 k )] ⊔ [( m − 4 , 7 + 6 k ) , ( m − 5 , 7 + 6 k ) , ( m − 6 , 7 + 6 k ) , . . . , (3 , 7 + 6 k )] ⊔ [(3 , 8 + 6 k ) , (2 , 8 + 6 k ) , (1 , 8 + 6 k ) , ( m, 8 + 6 k )] ⊔ [( m, 9 + 6 k ) , ( m − 1 , 9 + 6 k ) , ( m − 1 , 10 + 6 k ) , ( m − 2 , 10 + 6 k ) , ( m − 2 , 11 + 6 k ) , ( m − 3 , 11 + 6 k )]), and α 9 = ⊔ s − 3 2 k =0 ([( m, 6 + 6 k ) , ( m − 1 , 6 + 6 k ) , ( m − 1 , 7 + 6 k ) , ( m − 2 , 7 + 6 k ) , ( m − 2 , 8 + 6 k ) , ( m − 3 , 8 + 6 k ) , ( m − 3 , 9 + 6 k ) , ( m − 4 , 9 + 6 k )] ⊔ [( m − 4 , 10 + 6 k ) , ( m − 5 , 10 + 6 k ) , ( m − 6 , 10 + 6 k ) , . . . , (3 , 10 + 6 k )] ⊔ [(3 , 11 + 6 k ) , (2 , 11 + 6 k ) , (1 , 11 + 6 k ) , ( m, 11 + 6 k )]). 12 It follows that min- seed ( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 12 + 1. Case 3. r = 1. In this case, let S 1 , S 2 , S 3 and S 4 denote t he sets ∪ s − 1 j =0 { (1 , 3 + 3 j ) , (2 , 4 + 3 j ) , (3 , 3 + 3 j ) } , ∪ t − 2 i =0 { (4 + 4 i, 1) , (6 + 4 i, 1) , (6 + 4 i, 2) } , ∪ s − 1 j =0 ∪ t − 2 i =0 { (4 + 4 i, 5 + 3 j ) , (5 + 4 i, 3 + 3 j ) , (6 + 4 i, 5 + 3 j ) , (7 + 4 i, 3 + 3 j ) } and ∪ s − 1 j =0 { ( m − 1 , 5 + 3 j ) , ( m, 4 + 3 j ) } , resp ectiv ely . Let S = { (2 , 1) , (3 , 2) , ( m − 1 , 1) , ( m, 2) } ∪ S 1 ∪ S 2 ∪ S 3 ∪ S 4 . It is straigh tfo r ward to c hec k that the target set S can influence all vertic es in V ( G ) \ S b y using the convinc ed sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 (see Figure 10 in App endix for a graphical illustration of this con vinced sequence α ), where α 1 = ⊔ s − 1 j =0 [(1 , 4 + 3 j ) , (2 , 3 + 3 j )], α 2 = ⊔ t − 2 i =0 [(7 + 4 i, 1 ) , (7 + 4 i, 2 ) , (5 + 4 i, 1) , (5 + 4 i, 2) , (4 + 4 i, 2)], α 3 = ⊔ s − 1 j =0 ⊔ t − 2 i =0 [(4 + 4 i, 3 + 3 j ) , (5 + 4 i, 5 + 3 j ) , (6 + 4 i, 3 + 3 j ) , (7 + 4 i, 5 + 3 j )], α 4 = [( m − 1 , 2) , ( m, 1)], α 5 = [(3 , 1) , (2 , 2 ) , (1 , 2) , (1 , 1)], and α 6 = ⊔ s − 1 j =0 ([( m, 3 + 3 j ) , ( m − 1 , 3 + 3 j )] ⊔ [( m − 1 , 4 + 3 j ) , ( m − 2 , 4 + 3 j ) , ( m − 3 , 4 + 3 j ) , . . . , (3 , 4 + 3 j )] ⊔ [(3 , 5 + 3 j ) , (2 , 5 + 3 j ) , (1 , 5 + 3 j ) , ( m, 5 + 3 j )]). Therefore min-seed( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 12 + 1 4 . Case 4. r = 2 a nd s is ev en. In this case, let S 1 , S 2 , S 3 and S 4 denote the sets ∪ s − 1 j =0 { (1 , 3 + 3 j ) , (2 , 4 + 3 j ) , (3 , 3 + 3 j ) , (4 , 5 + 3 j ) , (5 , 3 + 3 j ) } , ∪ t − 3 i =0 { (6 + 4 i, 1) , (8 + 4 i, 1) , (8 + 4 i, 2) } , ∪ s − 1 j =0 ∪ t − 3 i =0 { (6 + 4 i, 5 + 3 j ) , (7 + 4 i, 3 + 3 j ) , (8 + 4 i, 5 + 3 j ) , (9 + 4 i, 3 + 3 j ) } a nd ∪ s − 1 j =0 { ( m − 4 , 5 + 3 j ) , ( m − 3 , 4 + 3 j ) , ( m − 2 , 3 + 3 j ) , ( m − 1 , 5 + 3 j ) , ( m, 4 + 3 j ) } , resp ectiv ely . Let S = { (2 , 1) , (3 , 2) , (4 , 1) , (5 , 2) , ( m − 4 , 1) , ( m − 3 , 2) , ( m − 1 , 1) , ( m, 2) } ∪ S 1 ∪ S 2 ∪ S 3 ∪ S 4 . It can readily b e c hec ke d that the target set S can influence a ll vertice s in V ( G ) \ S b y using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 ⊔ α 7 ⊔ α 8 ⊔ α 9 ⊔ α 10 (see Fig ure 11 in App endix for a graphical illustration of this conv inced sequence α ), where α 1 = ⊔ s − 1 j =0 [(1 , 4 + 3 j ) , (2 , 3 + 3 j )], α 2 = [(4 , 2) , (5 , 1 ) ], α 3 = ⊔ t − 3 i =0 [(9 + 4 i, 1 ) , (9 + 4 i, 2 ) , (7 + 4 i, 1) , (7 + 4 i, 2) , (6 + 4 i, 2)], α 4 = ⊔ s − 1 j =0 [(4 , 3 + 3 j ) , (5 , 5 + 3 j )], α 5 = ⊔ s − 1 j =0 ⊔ t − 3 i =0 [(6 + 4 i, 3 + 3 j ) , (7 + 4 i, 5 + 3 j ) , (8 + 4 i, 3 + 3 j ) , (9 + 4 i, 5 + 3 j )], 13 α 6 = [( m − 4 , 2) , ( m − 3 , 1 )], α 7 = ⊔ s − 2 2 k =0 ([( m − 3 , 3 + 6 k ) , ( m − 4 , 3 + 6 k )] ⊔ [( m − 4 , 4 + 6 k ) , ( m − 5 , 4 + 6 k ) , ( m − 6 , 4 + 6 k ) , . . . , (3 , 4 + 6 k )] ⊔ [(3 , 5 + 6 k ) , (2 , 5 + 6 k ) , (1 , 5 + 6 k ) , ( m, 5 + 6 k )] ⊔ [( m, 6 + 6 k ) , ( m − 1 , 6 + 6 k ) , ( m − 1 , 7 + 6 k ) , ( m − 2 , 7 + 6 k ) , ( m − 2 , 8 + 6 k ) , ( m − 3 , 8 + 6 k )]), α 8 = [( m − 2 , 1) , ( m − 2 , 2 ) , ( m − 1 , 2) , ( m, 1)], α 9 = [(3 , 1) , (2 , 2 ) , (1 , 2) , (1 , 1)], and α 10 = ⊔ s − 2 2 k =0 ([( m, 3 + 6 k ) , ( m − 1 , 3 + 6 k ) , ( m − 1 , 4 + 6 k ) , ( m − 2 , 4 + 6 k ) , ( m − 2 , 5 + 6 k ) , ( m − 3 , 5 + 6 k ) , ( m − 3 , 6 + 6 k ) , ( m − 4 , 6 + 6 k )] ⊔ [( m − 4 , 7 + 6 k ) , ( m − 5 , 7 + 6 k ) , ( m − 6 , 7 + 6 k ) , . . . , (3 , 7 + 6 k )] ⊔ [( 3 , 8+ 6 k ) , (2 , 8+ 6 k ) , (1 , 8+ 6 k ) , ( m, 8 +6 k )]). It follows that min- seed ( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 12 + 1 2 . Case 5. r = 2 and s is o dd. In this case, let S 1 , S 2 , S 3 and S 4 denote t he sets ∪ s − 1 j =0 { (1 , 3 + 3 j ) , (2 , 4 + 3 j ) , (3 , 3 + 3 j ) , (4 , 5 + 3 j ) , (5 , 3 + 3 j ) } , ∪ t − 3 i =0 { (6 + 4 i, 1) , (8 + 4 i, 1) , (8 + 4 i, 2) } , ∪ s − 1 j =0 ∪ t − 3 i =0 { (6 + 4 i, 5 + 3 j ) , (7 + 4 i, 3 + 3 j ) , (8 + 4 i, 5 + 3 j ) , (9 + 4 i, 3 + 3 j ) } and ∪ s − 1 j =0 { ( m − 4 , 5 + 3 j ) , ( m − 3 , 4 + 3 j ) , ( m − 2 , 3 + 3 j ) , ( m − 1 , 5 + 3 j ) , ( m, 4 + 3 j ) } , resp ectiv ely . Let S = { (2 , 1) , (3 , 2 ) , (4 , 1) , (5 , 2) , ( m − 4 , 1) , ( m − 3 , 2) , ( m − 2 , 1) , ( m − 1 , 1) , ( m, 2) } ∪ S 1 ∪ S 2 ∪ S 3 ∪ S 4 . It is straightforw ard to c hec k that the target set S can influence a ll v ertices in V ( G ) \ S by using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 ⊔ α 7 ⊔ α 8 ⊔ α 9 ⊔ α 10 ⊔ α 11 (see Figure 12 in App endix for a graphical illustration of this con vinced sequenc e α ), where α 1 = ⊔ s − 1 j =0 [(1 , 4 + 3 j ) , (2 , 3 + 3 j )], α 2 = [(4 , 2) , (5 , 1 ) ], α 3 = ⊔ t − 3 i =0 [(9 + 4 i, 1 ) , (9 + 4 i, 2 ) , (7 + 4 i, 1) , (7 + 4 i, 2) , (6 + 4 i, 2)], α 4 = ⊔ s − 1 j =0 [(4 , 3 + 3 j ) , (5 , 5 + 3 j )], α 5 = ⊔ s − 1 j =0 ⊔ t − 3 i =0 [(6 + 4 i, 3 + 3 j ) , (7 + 4 i, 5 + 3 j ) , (8 + 4 i, 3 + 3 j ) , (9 + 4 i, 5 + 3 j )], α 6 = [( m − 4 , 2) , ( m − 3 , 1 ) , ( m − 2 , 2) , ( m − 1 , 2) , ( m, 1)], α 7 = [(3 , 1) , (2 , 2 ) , (1 , 2) , (1 , 1)], α 8 = [( m − 3 , 3) , ( m − 4 , 3 )] ⊔ [( m − 4 , 4) , ( m − 5 , 4) , ( m − 6 , 4) , . . . , (3 , 4)] ⊔ [(3 , 5) , (2 , 5) , (1 , 5) , ( m, 5)], α 9 = [( m, 3) , ( m − 1 , 3) , ( m − 1 , 4) , ( m − 2 , 4) , ( m − 2 , 5) , ( m − 3 , 5)], 14 α 10 = ⊔ s − 3 2 k =0 ([( m − 3 , 6 + 6 k ) , ( m − 4 , 6 + 6 k )] ⊔ [( m − 4 , 7 + 6 k ) , ( m − 5 , 7 + 6 k ) , ( m − 6 , 7 + 6 k ) , . . . , (3 , 7 + 6 k )] ⊔ [(3 , 8 + 6 k ) , (2 , 8 + 6 k ) , (1 , 8 + 6 k ) , ( m, 8 + 6 k )] ⊔ [( m, 9 + 6 k ) , ( m − 1 , 9 + 6 k ) , ( m − 1 , 10 + 6 k ) , ( m − 2 , 10 + 6 k ) , ( m − 2 , 11 + 6 k ) , ( m − 3 , 11 + 6 k )]), and α 11 = ⊔ s − 3 2 k =0 ([( m, 6 + 6 k ) , ( m − 1 , 6 + 6 k ) , ( m − 1 , 7 + 6 k ) , ( m − 2 , 7 + 6 k ) , ( m − 2 , 8 + 6 k ) , ( m − 3 , 8 + 6 k ) , ( m − 3 , 9 + 6 k ) , ( m − 4 , 9 + 6 k )] ⊔ [( m − 4 , 10 + 6 k ) , ( m − 5 , 10 + 6 k ) , ( m − 6 , 10 + 6 k ) , . . . , (3 , 10 + 6 k )] ⊔ [(3 , 11 + 6 k ) , (2 , 11 + 6 k ) , (1 , 11 + 6 k ) , ( m, 11 + 6 k )]). W e conclude that min-seed( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 12 + 3 2 . Case 6. r = 3. In this case, let S 1 , S 2 , S 3 and S 4 denote t he sets ∪ s − 1 j =0 { (1 , 3 + 3 j ) , (2 , 4 + 3 j ) , (3 , 3 + 3 j ) , (4 , 5 + 3 j ) , (5 , 3 + 3 j ) } , ∪ t − 2 i =0 { (6 + 4 i, 1) , ( 8 + 4 i, 1) , (8 + 4 i, 2) } , ∪ s − 1 j =0 ∪ t − 2 i =0 { (6 + 4 i, 5 + 3 j ) , (7 + 4 i, 3 + 3 j ) , (8 + 4 i, 5 + 3 j ) , (9 + 4 i, 3 + 3 j ) } and ∪ s − 1 j =0 { ( m − 1 , 5 + 3 j ) , ( m, 4 + 3 j ) } , resp ective ly . Let S = { (2 , 1) , (3 , 2) , (4 , 1) , (5 , 2 ) , ( m − 1 , 1) , ( m, 2) } ∪ S 1 ∪ S 2 ∪ S 3 ∪ S 4 . It is straightforw ard to c hec k that the target set S can influence a ll v ertices in V ( G ) \ S by using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 ⊔ α 5 ⊔ α 6 ⊔ α 7 ⊔ α 8 (see Figure 13 in App endix for a gra phical illustration of this con vinced sequence α ), where α 1 = ⊔ s − 1 j =0 [(1 , 4 + 3 j ) , (2 , 3 + 3 j )], α 2 = [(4 , 2) , (5 , 1 ) ], α 3 = ⊔ t − 2 i =0 [(9 + 4 i, 1 ) , (9 + 4 i, 2 ) , (7 + 4 i, 1) , (7 + 4 i, 2) , (6 + 4 i, 2)], α 4 = ⊔ s − 1 j =0 [(4 , 3 + 3 j ) , (5 , 5 + 3 j )], α 5 = ⊔ s − 1 j =0 ⊔ t − 2 i =0 [(6 + 4 i, 3 + 3 j ) , (7 + 4 i, 5 + 3 j ) , (8 + 4 i, 3 + 3 j ) , (9 + 4 i, 5 + 3 j )], α 6 = [( m − 1 , 2) , ( m, 1)], α 7 = [(3 , 1) , (2 , 2 ) , (1 , 2) , (1 , 1)], and α 8 = ⊔ s − 1 j =0 ([( m, 3 + 3 j ) , ( m − 1 , 3 + 3 j )] ⊔ [( m − 1 , 4 + 3 j ) , ( m − 2 , 4 + 3 j ) , ( m − 3 , 4 + 3 j ) , . . . , (3 , 4 + 3 j )] ⊔ [(3 , 5 + 3 j ) , (2 , 5 + 3 j ) , (1 , 5 + 3 j ) , ( m, 5 + 3 j )]). Therefore min-seed( C m ⊘ C n , 3) ≤ | S | = mn 3 + m 12 + 3 4 . T his completes the pro of of the theorem. Theorem 12 If m ≡ 0 (mo d 3) and n ≥ 2 , then min-seed( C m ⊘ C n , 3) = mn 3 + 1 . 15 Pro of. Let G = C m ⊘ C n and m = 3 t . The pro of is divided in to tw o cases, according to the parity of n . Case 1. n is ev en. Denote b y S 1 , S 2 and S 3 the sets ∪ n − 2 2 j =0 { (1 , 1+ 2 j ) , (2 , 2+ 2 j ) } , ∪ t − 2 i =0 { (4 + 3 i, 2) , (6 + 3 i, 1) } and ∪ n − 4 2 j =0 ∪ t − 2 i =0 { (4 + 3 i, 4 + 2 j ) , (5 + 3 i, 3 + 2 j ) } , respective ly . Let S = S 1 ∪ S 2 ∪ S 3 ∪ { (3 , 1) } . It is straightforw ard to che ck that t he target set S can influence all v ertices in V ( G ) \ S b y using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 , where α 1 = ⊔ n − 2 2 j =0 [(2 , 1 + 2 j ) , (1 , 2 + 2 j )], α 2 = ⊔ t − 2 i =0 ⊔ n − 4 2 j =0 [(4 + 3 i, 3 + 2 j ) , (5 + 3 i, 4 + 2 j ) ], α 3 = [(3 , 2) , (3 , 3) , (3 , 4) , . . . , (3 , n )], and α 4 = ⊔ t − 2 i =0 ([(4 + 3 i, 1) , (5 + 3 i, 1) , (5 + 3 i, 2)] ⊔ [(6 + 3 i, 2) , (6 + 3 i, 3) , (6 + 3 i, 4) , . . . , (6 + 3 i, n )]) (see Figure 14 in App endix f or a graphical illustration of this convinc ed sequence α ). Therefore min-seed( C m ⊘ C n , 3) ≤ | S | = mn 3 + 1. By Theorem 2 (a), w e conclude that min-seed( C m ⊘ C n , 3) = mn 3 + 1. Case 2. n is o dd. By Theorem 8, it suffices t o consider only the case when n ≥ 5 . D enote b y S 1 , S 2 and S 3 the sets ∪ t − 1 i =0 { (1 + 3 i, 1) , (2 + 3 i, 2) , (3 + 3 i, 3) } , ∪ n − 5 2 j =0 { (1 , 5 + 2 j ) , (2 , 4 + 2 j ) } and ∪ n − 5 2 j =0 ∪ t − 2 i =0 { (4 + 3 i, 4 + 2 j ) , (5 + 3 i, 5 + 2 j ) } , respectiv ely . Let S = S 1 ∪ S 2 ∪ S 3 ∪ { (1 , 3) } . It is straightforw ard to che ck that t he target set S can influence a ll v ertices in V ( G ) \ S b y using the con vinced sequence α = α 1 ⊔ α 2 ⊔ α 3 ⊔ α 4 (see Figure 15 in App endix for a graphical illustration of this con vinced sequenc e α ), where α 1 = ⊔ n − 3 2 j =0 [(2 , 1 + 2 j ) , (1 , 2 + 2 j )], α 2 = ⊔ t − 2 i =0 ⊔ n − 7 2 j =0 [(4 + 3 i, 5 + 2 j ) , (5 + 3 i, 6 + 2 j )], α 3 = ⊔ t − 2 i =0 ([( m − 3 i, n ) , ( m − 3 i, n − 1) , . . . , ( m − 3 i, 4)] ⊔ [( m − 1 − 3 i, 4) , ( m − 1 − 3 i, 3) , ( m − 2 − 3 i, 3) , ( m − 2 − 3 i, 2) , ( m − 3 i, 2) , ( m − 3 i, 1) , ( m − 1 − 3 i, 1) , ( m − 2 − 3 i, n )]), and α 4 = [(3 , 2) , (3 , 1 ) , (2 , n )] ⊔ [(3 , n ) , (3 , n − 1) , . . . , (3 , 4)]. Therefore min-seed ( C m ⊘ C n , 3) ≤ | S | = mn 3 + 1, and hence b y Theorem 2(a) w e get min-seed( C m ⊘ C n , 3) = mn 3 + 1. This completes the pro of of t he theorem. References [1] E. Ac k erman, O. Ben-Zwi, G. 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W a tkins, A The or em o n T ait Colourings with an Applic ation to the Gener al- ize d Petersen Gr aphs , Jo urnal of Com binato r ial Theory , 6 (1969) 152- 164. 18 App endix : [Not for publication - for referees’ information only] Figure 1. min-seed( C 11 ⊘ C 3 , 3) = 12 (left) and min-seed( C 12 ⊘ C 3 , 3) = 13 ( righ t), where the targ et set S is the set of a ll blac k v ertices, and the con vinced sequence α 1 ⊔ α 2 ⊔ α 3 is illustrated b y three colored directed paths. Figure 2. min-seed( C 9 ⊘ C 9 , 3) = 28. 19 Figure 3. min-seed( C 12 ⊘ C 24 , 3) = 97. Figure 4. min-seed( C 12 ⊘ C 21 , 3) ≤ 86. 20 Figure 5. min-seed( C 9 ⊘ C 13 , 3) ≤ 41. Figure 6. min-seed( C 12 ⊘ C 22 , 3) ≤ 90. 21 Figure 7. min-seed( C 12 ⊘ C 25 , 3) ≤ 103. Figure 8. min-seed( C 16 ⊘ C 26 , 3) ≤ 140. 22 Figure 9. min-seed( C 16 ⊘ C 23 , 3) ≤ 125. Figure 10. min-seed( C 13 ⊘ C 20 , 3) ≤ 88. 23 Figure 11. min-seed( C 18 ⊘ C 26 , 3) ≤ 158. Figure 12. min-seed( C 18 ⊘ C 23 , 3) ≤ 141. 24 Figure 13. min-seed( C 15 ⊘ C 20 , 3) ≤ 102. Figure 14. min-seed( C 12 ⊘ C 14 , 3) = 57. 25 Figure 15. min-seed( C 12 ⊘ C 15 , 3) = 61. 26