Hamiltonian Paths in Two Classes of Grid Graphs

Hamiltonian P aths in Tw o Classes o f Grid Graphs F atemeh Kesha v arz-Kohjerdi a Dep artment of Computer Engine ering, Islamic Azad University, North T ehr an Br anch, T ehr an, Ir an. Alireza Bagheri b Dep artment of Computer Engine ering & IT, Amirka bir University of T e chnolo gy, T ehr an, Ir an. a Corr esp onding author: fatemeh.kesh avarz.2003@gmail.c om b ar b agheri@aut.ac. ir Abstract In this pap er, we giv e the neces sary and suffic ient conditions for the existence of Hamiltonian pa ths in L − alphab et and C − alphab et g r id graphs. W e also presen t a linear-time algorithm for finding Hamiltonian paths in t hese graphs. Key wor ds: Hamiltonian path, Hamiltonian cycle, G rid graph, Alphab et grid graph, Rectangular grid g r aph. AMS sub j ect classification: 05 C 45 1. In tro duction A Hamiltonian path in a gr a ph G ( V , E ) is a simple path t ha t includes ev ery v ertex in V . The problem of deciding whether a giv en graph G has a Hamiltonian path is a well-kno wn NP-complete pr o blem [2, 3]. Rectangular grid graphs first app eared in [5], where Luccio and Mugnia tried to solv e t he Hamiltonian pa th problem. The Hamiltonia n path problem w as studied fo r grid graphs in [4], where the author s g a ve the necessary and sufficien t condi- tions for the existence o f Hamiltonian paths in rectangular g rid graphs a nd pro ve d that the problem for general grid graphs is NP-complete. Also, the authors in [8] presen ted sufficien t conditions for a grid g raph to b e Hamilto- nian and prov ed that all finite grid graphs o f positive width hav e Hamilto nia n line g raphs. Chen et al. [1] improv ed the algorithm of [4 ] and presen ted a parallel algo r ithm for the problem in mesh arc hitecture. In [7, 6], Salman Pr eprint submitte d to Elsevier Novemb er 21, 2018 et al. determined the classes of alphab et graphs whic h con tain Hamilton cycles. In this paper, w e obtain the necessary and sufficien t conditions for a L − alphab et and C − alphab et graphs to ha ve Hamiltonian paths. Also, w e presen t a linear-time algorithm for finding Hamiltonian paths in these graphs. 2. Preliminaries In this section, w e presen t some definitions and previously established results on the Hamiltonian path problem in grid graphs whic h app eared in [1, 4, 6, 7]. The two-dimensional inte ger grid G ∞ is an infinite graph with the ve rtex set o f all the points of the Euclidean plane with in teger co ordinat es. In this graph, there is an edge b et w een any t w o v ertices of unit distance. F or a v ertex v of t his g raph, let v x and v y denote x and y co ordinates of its corresp onding p oint. W e color the vertice s of the t w o -dimensional integer grid b y black and white colors. A v ertex υ is colo r ed white if υ x + υ y is ev en, and is colored blac k otherwise. A grid gr aph G g is a finite ve rtex-induced subgraph of the tw o-dimensional inte g er grid. In a grid graph G g , eac h v ertex has degree a t most four. Clearly , there is no edge b et wee n an y t wo v ertices of t he same color. Therefore, G g is a bipartite graph. Note that an y cycle or path in a bipartite gra ph alternates b et w een black and white v ertices. A r e ctang ular grid gr aph R ( m, n ) (or R for short) is a grid gr a ph whose v ertex set is V ( R ) = { υ | 1 ≤ υ x ≤ m, 1 ≤ υ y ≤ n } . In the figures, w e assume that (1 , 1) is the co ordinat es of the v ertex in t he low er left corner. T he size of R ( m, n ) is defined to b e m × n . R ( m, n ) is called o dd-size d if m × n is o dd, and is called even-size d otherwise. R ( m, n ) is called a k-r e ctang le if n = k . The following lemma states a result ab out the Hamiltonicit y of eve n- sized rectangular graphs. Lemma 2.1. [1] R ( m, n ) has a Hamiltonian cycle if and only if it is even- size d and m, n > 1 . Tw o different v ertices υ and υ ′ in R ( m, n ) are called c ol o r-c omp atible if either b oth υ and υ ′ are white and R ( m, n ) is o dd-sized, or υ and υ ′ ha v e differen t colors and R ( m, n ) is ev en-sized. Without loss of generality , we as- sume s x ≤ t x . F or m, n ≥ 3, a L − alphab et graph L ( m, n ) (o r L for short) and a C − alphab et graph C ( m, n ) (or C f or short) are subgraphs of R (3 m − 2 , 5 n − 4) induced 2 b y V ( R ) \{ V | v x = m + 1 , . . . , 3 m − 2 and v y = n + 1 , . . . , 5 n − 4 } , and V ( R ) \{ V | v x = m + 1 , . . . , 3 m − 2 and v y = n + 1 , . . . , 4 n − 4 } , resp ectiv ely . These alphab et graphs ar e shown in Figure 1 for m = 4 and n = 3. An alphab et graph is called o dd-size d if its corresp onding rectangular graph is o dd-sized, and is called even-size d otherwise . In the follo wing by L ( m, n ) we mean an L − alphab et grid g r a ph L ( m, n ), and b y C ( m, n ) w e mean an C − alphab et g r id graph C ( m, n ). W e use P ( A ( m , n ) , s, t ) to indicate the problem of finding a Hamiltonian path b e- t w een v ertices s and t in grid graph A ( m, n ), and use ( A ( m, n ) , s, t ) to indi- cate the grid graph A ( m, n ) with t w o sp ecified distinct vertic es s and t o f it, where A is rectangular grid graph, L − alphab et graph or C − alphab et g r a ph. ( A ( m, n ) , s, t ) is Hamiltonian if there is a Hamiltonian path b etw een s and t in A ( m, n ). In this pap er, since ev en × odd L − alphab et ( C − alphab et) graph and odd × ev en L − alphab et graph a re isomorphic, then w e only consider ev en × odd L − alphab et ( C − alphab et) graphs. An ev en-sized grid graph contains the same num ber of blac k and white v er- tices. Hence, the t wo end-v ertices of an y Hamiltonian path in the gra ph m ust hav e differen t colors. Similarly , in a n o dd-sized g rid graph the n umber of white v ertices is one more than t he num b er of blac k v ertices . There- fore, the t w o end-v ertices of an y Hamiltonian path in suc h a graph m ust b e white. Hence, the color- compatibilit y of s and t is a necessary condition fo r ( R ( m, n ) , s, t ) to ha v e Hamilto nia n. F urthermore, Ita i et al. [4] sho w ed that if one of the followin g conditions hold, then ( R ( m, n ) , s, t ) is no t Hamiltonian: (F1) R ( m, n ) is a 1-rectangle and either s or t is not a corner v ertex (Figure 2(a)). (F2) R ( m, n ) is a 2-rectangle and ( s, t ) is a nonboundary edge, i.e. ( s, t ) is an edge and it is not on the o uter face (Figure 2(b)). (F3) R ( m, n ) is isomorphic to a 3-rectangle R ′ ( m, n ) suc h that s and t are mapp ed to s ′ and t ′ , and: 1. m is ev en, 2. s ′ is blac k, t ′ is white, 3. s ′ y = 2 and s ′ x < t ′ x (Figure 2(c)) or s ′ y 6 = 2 and s ′ x < t ′ x − 1 (Figure 2(d)). A Hamiltonian path problem P ( R ( m, n ) , s, t ) is ac c eptable if s and t are color-compatible and ( R, s, t ) do es not satisfy any of the conditions ( F 1), 3 ( F 2) and ( F 3). The follo wing theorem has b een pro ved in [4]. Theorem 2.1. L et R ( m, n ) b e a r e ctangular gr aph and s and t b e two distinct vertic es. Then ( R ( m, n ) , s, t ) is Hamiltonian if and only if P ( R ( m, n ) , s, t ) is ac c eptable. (a) (b) ()  - ? 6 2 m  2 4 n  4 6 ? -  n m  - ? 6 2 m  2 3 n  4 6 ? 6 ? n n  - m Figure 1: (a) A rectangular grid graph R (10 , 11), (b) a L − alphab et grid graph L (4 , 3), (c) a C − alphab et grid graph C (4 , 3). s s s s t t t t (a) (b) () (d) Figure 2: Rectangular grid graphs in whic h there is no Hamiltonian path b etw een s and t . 3. The Hamiltonian path in alphab et graphs In this section, w e g iv e sufficien t and necessary conditions fo r the exis- tence of a Hamiltonian path in L − alphab et and C − alphab et gra phs. W e also presen t an algorit hm for finding a Hamilto nia n path b etw een tw o given v ertices o f these graphs. A sep ar ation of L − alphab et g raph is a partition of L into t wo v ertex dis- join t rectangular grid graphs R 1 and R 2 , i.e. V ( L ) = V ( R 1 ) ∪ V ( R 2 ), and 4 V ( R 1 ) ∩ V ( R 2 ) = ∅ . A separ ation of C − alphab et graph is a partition of C in to a L − alphab et graph and a rectangular gr id gra ph. Lemma 3.1. L et A ( m, n ) b e a L − alphab et or C − alphab et grid gr aph and let R b e the smal lest r e ctangular grid gr a ph includes A . If ( A ( m, n ) , s, t ) has a Hamiltonian p ath b etwe en s and t , then ( R ( m, n ) , s, t ) also has a Hamiltonian p ath. Pr o of . Since R − A is a rectangular grid graph with ev en size of (2 m − 2) × (4 n − 4) or (2 m − 2) × ( 3 n − 4), then b y Lemma 2.1 it has a Hamilto- nian cycle. By combining the Hamiltonian cycle and t he Hamiltonian path of ( A ( m, n ) , s , t ) of [1], a Hamiltonian path b et w een s and t fo r ( R, s, t ) is obtained. Com bining Lemma 3.1 and Theorem 2 .1 the follo wing corollary is trivial. Corollary 3.1. L et A ( m, n ) b e a L − alphab et or C − alphab et grid gr aph and R b e its smal le st incl udin g r e ctangular grid gr aph. If ( A ( m, n ) , s, t ) has a Hamiltonian p ath b etwe en s and t , then s and t must b e c olor-c omp atible in R ( m, n ) . Therefore, the color-compatibility of s and t is a neces sary conditio n for ( L ( m, n ) , s, t ) and ( C ( m, n ) , s, t ) to hav e Hamiltonian paths. The length of a path in a grid graph means the num b er of v ertices of the path. In any grid graph, the length of an y path b etw een t w o same-colored v ertices is o dd and t he length of an y path b etw een t w o different-colored v ertices is ev en. Lemma 3.2. L et L ( m, n ) b e a L − alphab et grid gr aph and s a nd t b e two given vertic es of L . L et R (2 m − 2 , n ) and R ( m, 5 n − 4) b e a sep ar ation of L ( m, n ) . If t x > m + 1 and R ( 2 m − 2 , n ) satisfies c on dition ( F 3) , then L ( m, n ) do es not have any Hamiltonian p ath b etwe en s and t . Pr o of . Without loss of generality , let s and t b e color-compatible. Assume that R (2 m − 2 , n ) satisfies condition ( F 3) . W e sho w that there is no Hamilto- nian path in L ( m, n ) b etw ee n s a nd t . Assume to the con tra ry t ha t L ( m, n ) has a Hamiltonian path P . The follow ing cases are p ossible: Case 1. In this case, t is in R (2 m − 2 , n ) and s is not in R (2 m − 2 , n ). Since n = 3 there are exactly three v ertices v , w and u in R (2 m − 2 , n ) whic h are connected to R ( m, 5 n − 4), as shown in Figure 3. The followin g sub-cases 5 are p ossible for the Hamiltonian path P . Case 1.1. The Hamiltonian path P of L ( m, n ) that starts from s may en ter to R (2 m − 2 , n ) for t he first time through one of the v ertices v , w or u and passes throug h all the v ertices of R (2 m − 2 , n ) and end at t . This case is not p ossible b ecause we assumed that R (2 m − 2 , n ) satisfies ( F 3) ( t ′ = t and s ′ = w ). Case 1.2. The Hamiltonian path P of L ( m, n ) ma y en ter to R (2 m − 2 , n ), passes through s o me v ertices o f it, then lea v es it and en ter it again and passes through all the remaining v ertices of it and finally ends at t . In this case, t w o sub-paths of P whic h are in R (2 m − 2 , n ) are called P 1 and P 2 , P 1 from v to u ( v to w or u to w ) and P 2 from w to t ( u to t or v to t ). This case is not also p ossible b ecause the size of P 1 is o dd (ev en) and the size of P 2 is ev en (o dd), t hen | P 1 + P 2 | is o dd while R (2 m − 2 , n ) is ev en, whic h is a con tradiction. Case 2. In this case, s and t are in R (2 m − 2 , n ). The followin g cases ma y b e considere d: Case 2.1. The Hamiltonian path P of L ( m, n ) starts from s passes through some v ertices of R (2 m − 2 , n ), lea v es R (2 m − 2 , n ) at v ( u ), then passes thro ugh all the vertice s of R ( m, 5 n − 4) and ree nter t o R (2 m − 2 , n ) at w go es to u ( v ) and passes through all the remaining v ertices o f R (2 m − 2 , n ) and ends at t . In this case b y connecting v ( u ) to w we obtain a Hamiltonian path fr o m s to t in R (2 m − 2 , n ), whic h con tradicts to the assumption that R (2 m − 2 , n ) satisfies ( F 3), see Figure 4(a). Case 2.2. The Hamiltonian path P of L ( m, n ) starts from s leav es R (2 m − 2 , n ) at v ( u ), then passes through all the v ertices of R ( m, 5 n − 4) and reen ter to R (2 m − 2 , n ) at u ( v ) go es to w and passes throug h a ll the remaining v ertices of R (2 m − 2 , n ) and ends at t . In this case, t w o parts of P resides in R (2 m − 2 , n ). The part P 1 starts from s ends at v ( u ), and the part P 2 starts from u ( v ) ends at t . The size o f P 1 is ev en and the size of P 2 is o dd while the size of R ( 2 m − 2 , n ) is eve n, whic h is a contradiction, see Figure 4(b). Case 2.3. Another case that ma y imagine is that the Hamiltonian path P of L ( m, n ) starts from s lea v es R (2 m − 2 , n ) at w and reen ters R (2 m − 2 , n ) at v ( u ) and then go es to t . But in this case v ertex u ( v ) can not b e in P , whic h is a contradiction, see Figure 4(c). Th us the pro of of Lemma 3.2 is completed. 6 v w u  -  - ? 6 ? 6 m n 2 m  2 5 n  4 Figure 3: v w u v w u v w u s t s t s t (a) (b)  Figure 4: L − alphab et grid graphs in wh ic h there is no Hamiltonian p ath b et w een s and t . Lemma 3.3. Assume that C ( m, n ) is a C − alphab et grid gr aph and s and t ar e two given vertic e s of C ( m, n ) . L et L ( m , n ) and R (2 m − 2 , n ) b e a sep ar ation of C ( m, n ) . If L ( m, n ) d o es not have Hamiltonian p ath, then C ( m, n ) do es not have Hamiltonian p ath b etwe en s and t . Pr o of . The proo f is similar t o the pro of of Lemma 3.2 , for more details see Figure 5. A Hamiltonian path problem P ( L ( m, n ) , s, t ) is ac c eptable if s and t are color-compatible and R (2 m − 2 , n ) do es not satisfy the condition ( F 3), and also P ( C ( m, n ) , s, t ) is ac c eptable if P ( L ( m, n ) , s, t ) is acceptable, where L ( m, n ) is a partition of C ( m, n ). No w, w e are ha v e sho wn that all acce ptable Hamiltonian path problems hav e 7 s t s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . .. . .. .. . .. .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . .. . .. .. . .. (a) (b) () t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. .. . .. . .. .. . .. . .. .. . .. .. . .. . .. .. . .. . . . Figure 5: C − alphab et grid graph s in wh ic h there is no Hamiltonian path b et wee n s and t . solutions. Our algorithm is described in the follow ing: If the giv en graph is L − alphab et, then it is divided into tw o rec t a ngular grid graphs and t w o Hamiltonian paths in them are found by the a lgorithm of [1]. If the giv en graph is C − alphab et, then it is divided in to a L − alphab et graph and a rectangular grid graph, and the Hamiltonian path in L − alphab et graph is found as b efore. In the follow ing w e discuss the details of this dividing and merging. A rectangular subgraph S of L − alphab et or C − alphab et graph A strips a Hamiltonian path problem P ( A ( m, n ) , s, t ), if: 1. S is ev en-sized. 2. S and A − S is a separation of A . 3. s, t ∈ A − S 4. A − S is acceptable. Lemma 3.4. L et P ( L ( m, n ) , s, t ) b e an ac c eptable Hamiltonian p ath pr oblem, and S strips it. I f L − S has a Hamiltonian p ath b etwe en s and t , then ( L ( m, n ) , s, t ) has a Hamiltonian p ath b etwe en s and t . Pr o of . Assume that L − S has a Hamiltonian path H . S is an ev en-sized rectangular g r id graph and it ha s Hamiltonian cycle by Lemma 2.1. There exists a n edge ab ∈ H suc h that ab is on the b oundary of L − S facing S . A Hamiltonian path for ( L ( m, n ) , s, t ) can b e obtained b y merging H and the Hamiltonian cycle of S as sho wn in Figure 6(a). 8 Let ( R p , R q ) b e a separation of L ( m, n ). If s and t are in differen t par t i- tions, then w e consider t w o v ertices p a nd q suc h tha t s, p ∈ R p , q , t ∈ R q and ( R p , R q ) are acceptable. Therefore, a Hamiltonian path for ( L ( m, n ) , s, t ) can b e obtained by connecting t w o vertice s p and q as shown in Figure 7(a). . .. . .. . .. .. . .. . .. .. . .. . .. .. . .. .. . L  S S s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s t S L  S a b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . .. . .. .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. .. . .. . C  S S s t s t a b (a) (b) C  S . .. . .. .. . .. .. . .. . .. .. . .. . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S Figure 6: (a) A strip of L (3 , 3), (b) A strip of C (3 , 3). Lemma 3.5. L et P ( C ( m, n ) , s, t ) b e an ac c eptable Hamiltonian p ath p r oblem, and S s trips it. If C − S h a s a H a m iltonian p ath b etwe en s and t , then ( C ( m, n ) , s, t ) has a Ham iltonian p a th b etwe en s and t . Pr o of . The pro o f is similar to L emma 3.4. Notice that C − S is a L − alphab et grid graph, see Figure 6(b). Let ( R p , L q ) b e a separation of C ( m, n ). If s and t are in differen t parti- tions, then w e conside r tw o vertice s p a nd q suc h that s, p ∈ R p , q , t ∈ L q and ( R p , L q ) are acceptable. Therefore, a Hamiltonian path for ( L ( m, n ) , s, t ) can 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R p R q s t p q R P s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R q p q (a) t s s t L q R p . .. . .. . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p q . . .. .. . .. . .. .. . .. L q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. .. . .. . .. .. . . . . . . . . . . . . . . . R p p q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) . . . . . . . . . . . . . . . . . . . . . . . . Figure 7: (a) A sp lit of L (3 , 3), (b) A split of C (3 , 3). b e obtained by connecting t w o vertice s p and q as shown in Figure 7(b). F rom Corolla ry 3.1 and Lemmas 3.1, 3.2, 3.3, 3.4 and 3.5, the follo wing theorem holds. Theorem 3.1. L et A ( m, n ) b e a L − alphab et or C − alphab et gr ap h, and let s and t b e two distinc t vertic es of it. A ( m, n ) has a Hamiltonian p ath if and only if P ( A ( m, n ) , s, t ) is ac c eptable. Theorem 3.1 pro vides the necessary and sufficien t conditions for the ex- istence of Hamiltonian paths in L − alphab et and C − alphab et grid gra phs. Theorem 3.2. In L − alphab et and C − alphab et grid gr a p hs, a Ham iltonian p ath b etwe en any two vertic es s and t c an b e found in line ar time. 10 Pr o of . W e divide the pro blem in to t w o (or three) rectangular grid graphs in O(1). Then w e solv e the subproblem s in linear- time and merge the results in O(1) using the metho d prop osed in [1]. 4. Conclusion and future w ork W e presen ted a linear-time algorithm for finding a Hamiltonian path in L − alphab et and C − alphab et gr id graphs b etw een an y t wo g iven v ertices. Since the Hamilto nian path problem is NP-complete in general grid graphs, it remains op en if the problem is p olynomially solv able in solid grid graphs. References [1] S. D. Chen, H. Shen and R. T op or, An effic ient algorithm for construct- ing Ha miltonian paths in meshes, J. P arallel Computing, 28(9):1293- 1305, 2002. [2] R. Diestel, Graph Theory , Springer, New Y ork, 20 00. [3] M. R. Garey and D . S. Johnson, Computers and In tractability: A Guid to the Theory of NP-completeness, F reeman, San F rancisco, 197 9 . [4] A. Ita i, C. P apadimitriou and J. Szw arcfiter, Hamiltonian paths in g rid graphs, SIAM J. Comput., 1 1(4):676- 6 86, 1982 . [5] F. Luccio and C. Mugnia, Hamiltonian paths on a rectangular c hess- b oard, Pro c. 1 6th Ann ual Allerton Conference, 161-173 , 197 8 . [6] A. N. M. Salman, E. T. Bask oro and H. J. Bro ersma, Spanning 2- connected subgraphs in alphab et graphs, sp ecial class of g rid graphs, Pro c. ITB Sains & T ek, 35 A(1): 65- 70, 2003. [7] A. N. M. Salman, H. J. Bro ersma and C. A. Ro dger, More on spanning 2-connected subgraphs of alphab et graphs, sp ecial classes of grid graphs, Bulletin of the Institute of Com binatorics and Its Applic at io n, 45: 17-3 2 , 2005. [8] C. Zamfirescu and T. Z a mfirescu, Hamiltonian Prop erties of Grid Graphs, SIAM J. Math., 45 ( 4 ): 564- 570, 1992. 11