The Hamiltonian properties of supergrid graphs

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1 *Mauscript (PDF) Abstract The Hailtoia properties of supergrid graphs Ruo-Wei Hug, Chih-Chia Yao, ad Shag-Ju Cha Departet of Coputer Sciece ad Iforatio Egieerig, Chaoyag Uiversity of Techology, Wufeg, Taichug, Taiwa I this paper, we first itroduce a ovel class of graphs, aely supergrid. Supergrid graphs iclude grid graphs ad triagular grid graphs as their subgraphs. The Hailtoia cycle ad path probles for grid graphs ad triagular grid graphs were kow to be NP-coplete. However, they are ukow for supergrid graphs. The Hailtoia cycle (path) proble o supergrid graphs ca be applied to cotrol the stitchig traces of coputerized sewig achies. I this paper, we will prove that the Hailtoia cycle proble for supergrid graphs is NP-coplete. It is easily derived fro the Hailtoia cycle result that the Hailtoia path proble o supergrid graphs is also NP-coplete. We the show that two subclasses of supergrid graphs, icludig rectagular (parallelis) ad alphabet, always cotai Hailtoia cycles. Keywords: Hailtoia properties, supergrid graph, rectagular supergrid graph, alphabet supergrid graph, grid graph, triagular grid graph, coputerized sewig achie 1. Itroductio A Hailtoia cycle i a graph is a siple cycle i which each vertex of the graph appears exactly oce. A Hailtoia path i a graph is a siple path with the sae property. The Hailtoia cycle (resp., path) proble ivolves testig whether or ot a graph cotais a Hailtoia cycle (resp., path). A graph is said to be Hailtoia if it cotais a Hailtoia cycle. The Hailtoia probles iclude Hailtoia cycle ad Hailtoia path probles. They have uerous applicatios i differet areas, icludig establishig trasport routes, productio lauchig, the o-lie optiizatio of flexible aufacturig systes [1], coputig the perceptual boudaries of dot patters [0], patter recogitio [, 1, ], ad DNA physical appig [1]. It is well kow that the Hailtoia probles are NP-coplete for geeral graphs [, 0]. The sae holds true for bipartite graphs [], split graphs [], circle graphs [], udirected path graphs [], grid graphs [], ad triagular grid graphs [1]. I this paper, we will study the Hailtoia probles o supergrid graphs which cotai grid graphs ad triagular grid graphs as subgraphs. The two-diesioal iteger grid G is a ifiite graph whose vertex set cosists of all poits of the Euclidea plae with iteger coordiates ad i which two vertices are adjacet if ad oly if the (Euclidea) distace betwee the is equal to 1. A grid graph is a fiite, vertex-iduced subgraph of G. For a ode v i the plae with iteger coordiates, let v x ad v y represet the x ad y coordiates of ode v, respectively, deoted by v=(v x, v y ). If v is a vertex i a grid graph, the its possible adjacet vertices iclude (v x, v y 1), (v x 1, v y ), (v x + 1, v y ), ad (v x, v y + 1). For exaple, Fig. 1 depicts a fraget of graph G ad Fig. shows a grid graph. Recetly, the properties of triagular grid graphs, which cotai grid graphs as subgraphs, have received uch attetio. The two-diesioal triagular grid T is a ifiite graph obtaied fro G by addig all edges o the lies traced fro up-left to dow-right. A triagular grid graph is a fiite, vertex-iduced subgraph of T. The possible adjacet vertices of a vertex v=(v x, v y ) i a triagular grid graph cotai (v x, v y 1), (v x 1, v y ), (v x + 1, v y ), (v x, v y + 1), (v x 1, v y 1), ad (v x + 1, v y + 1). For istace, Fig. 1 depicts a fraget of graph T ad Fig. shows a triagular grid Correspodig author. Eail addresses: rwhug@cyut.edu.tw (Ruo-Wei Hug), ccyao@cyut.edu.tw (Chih-Chia Yao) Preprit subitted to Theoretical Coputer Sciece July 1, 01

2 ( vx, vy 1) ( v 1, v ) ( vx, vy) ( v +1, v ) x y ( vx, vy+ 1) (c) x y Fig. 1: A fraget of ifiite graph G, T, ad (c) S. (c) Fig. : A grid graph, a triagular grid graph, ad (c) a supergrid graph, where solid lies idicate the edges of the graphs. graph. Note that T is isoorphic to the origial ifiite triagular grid graph i the literature [1] but these graphs are differet whe cosidered as geoetric graphs. By the sae costructio of triagular grid graphs fro grid graphs, we propose a ew class of graphs, aely supergrid graphs, as follows. The two-diesioal supergrid S is a ifiite graph obtaied fro T by addig all edges o the lies traced fro up-right to dow-left. A supergrid graph is a fiite, vertex-iduced subgraph of S. The possible adjacet vertices of a vertex v=(v x, v y ) i a supergrid graph cotai (v x, v y 1), (v x 1, v y ), (v x + 1, v y ), (v x, v y + 1), (v x 1, v y 1), (v x + 1, v y + 1), (v x + 1, v y 1), ad (v x 1, v y + 1). Thus, supergrid graphs cotai grid graphs ad triagular grid graphs as subgraphs. For exaple, Fig. 1(c) depicts a fraget of graph S ad Fig. (c) shows a supergrid graph. Obviously, all grid graphs are bipartite [] but triagular grid graphs ad supergrid graphs are ot bipartite. The Hailtoia cycle ad path probles o grid graphs ad triagular grid graphs have bee show to be NP-coplete [1, ]. However, they are ukow for supergrid graphs. I this paper, we will prove that the Hailtoia cycle ad path probles for supergrid graphs are NP-coplete. The possible applicatio for the Hailtoia cycle (path) proble o supergrid graphs is preseted as follows. Cosider a coputerized sewig achie give a iage. The coputerized sewig software is used to copute the sewig traces of a coputerized sewig achie. There ay be two parts i a coputerized sewig software. The first part is to do iage processig for the iput iage, e.g. reduce order of colors ad iage thiig. It the produces soe sets of lattices i which every set of lattices represets a color i the iput iage for sewig. The secod part is give by a set of lattices ad the coputes a cycle (path) to visit the lattices of the set such that each lattice is visited exactly oce. Fially, the software trasits the stitchig trace of the coputed cycle (path) to the coputerized sewig achie, ad the achie the perfors the sewig work alog the trace o the object, e.g. clothes. For exaple, give a iage i Fig., the software first aalyzes the iage ad the produces seve colors of regios i which each regio is filled with the sae color ad ay cosist of soe discoected blocks, as show i Fig.. It the produces seve sets of lattices i which every set of lattices represets a regio, where each regio is filled by a sewig trace with the sae color ad it ay be partitioed ito ay o-cotiguous blocks. Fig.

3 (c) (d) Fig. : A iput iage for the coputerized sewig software, seve colors of regios produced by iage processig, (c) a set of lattices for oe regio of color, (d) a possible sewig trace for the set of lattices i (c), ad (e) a overview after coputig sewig traces of all regios of colors. (c) shows a set of lattices for oe regio of color, ad the software the coputes a sewig trace for the set of lattices, as depicted i Fig. (d). Sice each stitch positio of a sewig achie ca be oved to its eight eighbor positios (left, right, up, dow, up-left, up-right, dow-left, ad dow-right), oe set of lattices fors a supergrid graph which ay be discoected. Note that each lattice will be represeted by a vertex of a supergrid graph, each regio ay be separated ito ay blocks i which each block represeted a coected supergrid graph. The desired sewig trace of each set of adjacet lattices is the Hailtoia cycle (path) of the correspodig coected supergrid graph whe it is Hailtoia. Note that if the correspodig supergrid graph is ot Hailtoia, the the sewig trace cotais ore tha oe paths ad these paths ust be cocateated. After coputig the sewig traces of all regios of colors, the software the trasits the coputed stitchig trace to the coputerized sewig achie. Fig. (e) depicts the possible sewig result for the iage i Fig.. I additio, the structure of supergrid graphs ca be used to desig the etwork topology, ad its etwork diaeter is saller tha that of grid graphs. Related areas of ivestigatio are suarized as follows. Itai et al. [] showed that the Hailtoia cycle ad Hailtoia path probles for grid graphs are NP-coplete. They also gave the ecessary ad sufficiet coditios for a rectagular grid graph havig a Hailtoia path betwee two give vertices. Zafirescu et al. [] gave the (e)

4 sufficiet coditios for a grid graph havig a Hailtoia cycle, ad proved that all grid graphs of positive width have Hailtoia lie graphs. Later, Che et al. [] iproved the Hailtoia path algorith of [] o rectagular grid graphs ad preseted a parallel algorith for the Hailtoia path proble with two give edpoits i rectagular grid graph (esh). Also there is a polyoial-tie algorith for fidig Hailtoia cycles i solid grid graphs []. I [], Sala itroduced alphabet grid graphs ad deteried classes of alphabet grid graphs which cotai Hailtoia cycles. Keshavarz-Kohjerdi ad Bagheri [1] gave the ecessary ad sufficiet coditios for the existece of Hailtoia paths i alphabet grid graphs, ad preseted liear-tie algoriths for fidig Hailtoia paths with two give edpoits i these graphs. Recetly, Keshavarz-Kohjerdi et al. [] preseted a liear-tie algorith for coputig the logest path betwee two give vertices i rectagular grid graphs. The Hailtoia cycle (path) o triagular grid graphs has bee show to be NP-coplete [1]. Recetly, Reay ad Zafirescu [] proved that all -coected, liear-covex triagular grid graphs except oe special case cotai Hailtoia cycles. They also proved that all coected, locally coected triagular grid graphs (with oe exceptio) cotai Hailtoia cycles. I additio, the Hailtoia cycle proble o hexagoal grid graphs was kow to be NP-coplete []. For ore related works, we refer readers to [,,, 1, 1, 1,,,,,, ]. The rest of the paper is orgaized as follows. I Sectio, soe otatios ad basic teriologies are itroduced. Sectio shows that the Hailtoia cycle ad Hailtoia path probles for supergrid graphs are NP-coplete. I Sectio, we show that rectagular (parallelis) ad alphabet supergrid graphs are Hailtoia. Fially, we ake soe cocludig rearks i Sectio.. Notatios ad teriologies I this sectio, we will itroduce fudaetal teriologies ad sybols used i the paper. For graph-theoretic teriology ot defied i this paper, the reader is referred to []. Let G = (V, E) be a graph with vertex set V(G) ad edge set E(G). Let S be a subset of vertices i G, ad let u, v be two vertices i G. We write G[S ] for the subgraph of G iduced by S, G S for the subgraph G[V S ], i.e., the subgraph iduced by V S. I geeral, we write G v istead of G {v}. If (u, v) is a edge i G, we say that u is adjacet to v ad u, v are icidet to edge (u, v). The otatio u v (resp., u v) eas that vertices u ad v are adjacet (resp., o-adjacet). A eighbor of v i G is ay vertex that is adjacet to v. We use N G (v) to deote the set of eighbors of v i G. The subscript G of N G (v) ca be reoved fro the otatio if it has o abiguity. The degree of vertex v is the uber of vertices adjacet to the vertex v. The distace betwee u ad v is the legth of the shortest path betwee these two vertices. A path P of legth P 1 i a graph G, deoted by v 1 v v P 1 v P, is a sequece (v 1, v,, v P 1, v P ) of vertices such that (v i, v i+1 ) E(G) for 1 i< P. The first ad last vertices visited by path P are deoted by start(p) ad ed(p), respectively. We will use v i P to deote P visits vertex v i ad use (v i, v i+1 ) P to deote P visits edge (v i, v i+1 ). A path fro vertex v 1 to vertex v k is deoted by (v 1, v k )-path. I additio, we use P to refer to the set of vertices visited by path P if it is uderstood without abiguity. O the other had, a path is called the reversed path, deoted by rev(p), of path P if it visits the vertices of P fro ed(p) to start(p) i proper sequece; that is, the reversed path rev(p) of P=v 1 v v P 1 v P is v P v P 1 v v 1. A cycle is a path C such that V(C) ad start(c) ed(c). Let S be the ifiite graph whose vertex set cosists of all poits of the plae with iteger coordiates ad i which two vertices are adjacet if ad oly if the differece of their x or y coordiates is ot larger tha 1. A supergrid graph is a fiite, vertex-iduced subgraph of S. For a vertex v i a supergrid graph, let v x ad v y deote respectively x ad y coordiates of its correspodig poit. We color vertex v to be white if v x + v y 0 (od ); otherwise, v is colored to be black. The there are eight possible eighbors of vertex v icludig four white vertices ad four black vertices. Obviously, all grid graphs are bipartite [] but supergrid graphs are ot bipartite. Rectagular grid graphs first appeared i [], where Luccio ad Mugia tried to solve the Hailtoia path proble o the. Itai et al. [] gave ecessary ad sufficiet coditios for the existece of Hailtoia (s, t)-path i rectagular grid graphs, where s, t are two give vertices. I this paper, we expad the to a subclass of supergrid graphs, aely rectagular supergrid graphs. Let R(, ) be the supergrid graph whose vertex set V(R(, )) ={v = (v x, v y ) 1 v x ad 1 v y }. The, R(, ) cotais colus ad rows of vertices i S. A rectagular supergrid graph is a supergrid graph that is isoorphic to R(, ) for, 1. Thus ad, the diesios, specify a rectagular supergrid graph up to isoorphis. The size of R(, ) is defied to be, ad R(, ) is called -rectagle. The edge i the boudary of R(, ) is called boudary edge. For exaple, Fig.

5 = (1, 1) = (d) boudary edges (1, 1) (, ) (, ) ( 1, ) (0, ) (1, 1) (e) Fig. : Rectagular, parallelis, ad alphabet supergrid graphs, where a rectagular supergrid graph R(, ), two parallelis supergrid graphs P(, ) ad P(, ), (c) a L-alphabet supergrid graph L(, ), (d) a C-alphabet supergrid graph C(, ), (e) a F-alphabet supergrid graph F(, ), ad (f) a E-alphabet supergrid graph E(, ). shows a rectagular supergrid graph R(, ) which is called -rectagle ad cotais ( + ) = boudary edges. I the figures, we assue that (1, 1) is the coordiates of the up-left vertex, i.e. the leftost vertex of the first row, i a supergrid graph. A parallelis supergrid graph is defied siilar to R(, ). Let P(, ) be the supergrid graph with whose vertex set V(P(, ))={v = (v x, v y ) 1 v y ad v y v x v y + 1} or {v=(v x, v y ) 1 v y ad v y v x v y + 1}. A parallelis supergrid graph is a supergrid graph which is isoorphic to P(, ). For exaple, Fig. depicts two parallelis supergrid graphs P(, ) ad P(, ). I the above defiitio, there are two types of parallelis supergrid graphs. We ca see that they are isoorphic although they are differet whe cosidered as geoetric graphs. Note that the boudary edges of R(, ) ad P(, ) for a rectagle ad a parallelogra, respectively. I [], Sala first itroduced alphabet grid graphs, which for a subclass of grid graphs, ad studied soe properties of these graphs. Recetly, Keshavarz-Kohjerdi ad Bagheri [1] deteried the ecessary ad sufficiet coditios for the existece of Hailtoia (s, t)-path i alphabet grid graphs, where s, t are two give vertices. I this paper, we exted the to for a subclass of supergrid graphs, aely alphabet supergrid graphs. A alphabet supergrid graph is a fiite vertex-iduced subgraph of the rectagular supergrid graph of a certai type, as follows. For,, a L-alphabet supergrid graph L(, ), C-alphabet supergrid graph C(, ), F-alphabet supergrid graph F(, ), ad E-alphabet supergrid graph E(, ) are subgraphs of R(, ). These alphabet supergrid graphs are defied as show i Fig. (c) (f), where = ad =.. NP-copleteess I this sectio, we will prove that the Hailtoia cycle (path) proble for supergrid graphs is NP-coplete. I [] ad [1], the authors showed the Hailtoia cycle proble for grid graphs ad triagular grid graphs to be P(, ) P(, ) (c) (f)

6 x 1 x x x V 0 V 1 y 1 x 1 y 1 y y eb Fig. : A plaar bipartite graph B with axiu degree, ad a parity-preservig ebeddig eb(b) of B, where solid lies idicate the edges i the ebeddig paths. NP-coplete. We apply the idea of these proofs to show that the Hailtoia cycle proble reais NP-coplete for supergrid graphs. By usig siilar arguets, we ca prove that the Hailtoia path proble o supergrid graphs is still NP-coplete. Notice that grid graphs ad triagular grid graphs are ot subclasses of supergrid graphs; these classes of graphs have coo eleets (vertices) but i geeral they are distict. To prove the Hailtoia cycle proble o supergrid graphs to be NP-coplete, we establish a polyoial-tie reductio fro the Hailtoia cycle proble for plaar bipartite graphs with axiu degree. The followig theore was give i []. Theore.1. (See [].) The Hailtoia cycle proble for plaar bipartite graphs with axiu degree is NP-coplete. Let B=(V 0 V 1, E) be a plaar bipartite graph with axiu degree ad let G 1 be a rectagular supergrid graph, where V 0 ad V 1 are the bipartitio sets of B i which every edge jois a vertex i V 0 ad a vertex i V 1. Siilarly to the parity-preservig ebeddig [] of a bipartite graph ito a rectagular grid graph, let us itroduce the followig parity-preservig ebeddig eb of B ito G 1 (a oe-to-oe fuctio fro V 0 V 1 to the vertices of G 1 ad fro E to paths i G 1 ): 1. The vertices of V 0 are apped to white vertices of G 1, i.e., if v V 0, the eb(v) is colored by white.. The vertices of V 1 are apped to black vertices of G 1, i.e., if v V 1, the eb(v) is colored by black.. The edges of B are apped to vertex-disjoit paths of G 1, i.e., if e=(u, v) E, the eb(e) is a path P fro eb(u) to eb(v), ad the iterediate vertices of P do ot belog to ay other path. For exaple, Fig. shows a plaar bipartite graph B with axiu degree, ad a parity-preservig ebeddig eb(b) of B is depicted i Fig.. The followig lea is give i [] ad shows that the above parity-preservig ebeddig ca be doe i polyoial tie. Lea.. (See [].) Let B be a plaar bipartite graph with vertices ad axiu degree. The, a paritypreservig ebeddig eb(b) of B ito a rectagular grid (supergrid) graph R(k, k) ca be doe i polyoial tie, where k is a costat. Now give a plaar bipartite graph B with vertices ad axiu degree, we shall costruct a supergrid graph G s such that B has a Hailtoia cycle if ad oly if G s cotais a Hailtoia cycle. Let B=(V 0 V 1, E). The costructio of G s fro B is sketched as follows. First, we ebed graph B ito a rectagular supergrid graph R(k, k) for soe costat k, as described i Lea.. Let the ebeddig supergrid graph be G 1. I the secod step, we elarge the supergrid graph G 1 such that each edge i G 1 is trasfored ito a path with edges. Let the elarged supergrid graph be G. For exaple, Fig. shows the supergrid graph G elarged fro the supergrid graph G 1 i Fig. for the plaar bipartite graph B i Fig.. I the third step, each vertex of graph B is trasfored ito a cluster which is a sall supergrid graph. The vertex of B is called critical vertex i G. Fially, each path i G is siulated by a tetacle which is a series of -rectagles, ad the resultat graph is a supergrid graph G s. Now, we itroduce clusters ad tetacles as follows. We trasfor each vertex of V 0 ito a white cluster ad each vertex of V 1 ito a black cluster. A white cluster is a supergrid graph with vertices ad a black cluster is also a supergrid graph with 1 vertices. Fig. shows the white ad black clusters. The ceter of a white cluster (resp., black x y y

7 x 1 y 1 x x Fig. : The elarged supergrid graph G fro the ebeddig supergrid graph G 1 i Fig., where solid lies idicate the edges i G. cluster) is a white critical vertex (resp., black critical vertex). For exaple, critical vertex x (resp., y) is the ceter of a white cluster (resp., a black cluster) i Fig. (resp., Fig. ). The distace betwee ay vertex ad ceter i a cluster is at ost. The vertices a 1, a, a, a (resp., b 1, b, b, b ) i Fig. (resp., Fig. ) are called the corer vertices of cluster, ad the edges e 1, e, e, e i Fig. are called critical edges. I our costructio, the corer vertices together with critical edges of a cluster are used to coect to the other cluster. The followig two propositios show the properties of clusters. Propositio.. Let C be a white cluster, a 1, a, a, a be its corer vertices, ad let e 1, e, e, e be its critical edges, as i Fig.. The for 1 i< j, there exists a Hailtoia (a i, a j )-path of C which cotais all four critical edges{e 1, e, e, e }. Proof. By ispectio, the lea ca be verified. For exaple, Fig. ad Fig. depict a Hailtoia (a 1, a )- path ad a Hailtoia (a 1, a )-path of C, respectively, which cotai all critical edges{e 1, e, e, e }. Propositio.. Let C 1 be a black cluster, b 1, b, b, b be its corer vertices, ad let e 1, e, e, e be its critical edges, as i Fig.. The for 1 i< j, there exists a Hailtoia (b i, b j )-path of C 1 which cotais all four critical edges{e 1, e, e, e }. y y

8 e a e a 1 a x e 1 e a Fig. : A white cluster C, ad a black cluster C 1, where the ceters of C ad C 1 are critical vertices x ad y, respectively, the critical edges iclude e 1, e, e, e, ad the corer vertices iclude a 1, a, a, a or b 1, b, b, b. e a 1 a x a e e a b 1 b y b e b (c) e 1 e 1 e e b e e e b 1 b b 1 b y b e a 1 a x a Fig. : A Hailtoia (a 1, a )-path of a white cluster C, a Hailtoia (a 1, a )-path of a white cluster, (c) a Hailtoia (b 1, b )-path of a black cluster C 1, ad (d) a Hailtoia (b 1, b )-path of a black cluster, where arrow lies idicate the edges i such a path ad each critical edge is cotaied i the Hailtoia path. e e a b (d) y e 1 e 1 e 1 e e e b

9 a b c d a 1 b 1 a = c d 1 a 1 b b = c c d Fig. : A strip S (a, b; c, d) with corers a, b, c, d, ad a square tetacle T (a 1, b 1 ; c, d ) cosistig of strips. Proof. By ispectio. For exaple, For exaple, Fig. (c) ad Fig. (d) depict a Hailtoia (b 1, b )-path ad a Hailtoia (b 1, b )-path of C 1, respectively, which cotai all critical edges{e 1, e, e, e }. I our costructio, the path i G is siulated by a series of -rectagles, called tetacle. We will use the siilar techique i [] to ake it. A strip is a rectagular supergrid graph with at least squares (see Fig. ), i.e., it is isoorphic to R(, ) with. The strip with corers a, b, c, d (the degree of every corer is ), as i Fig., is deoted by S (a, b; c, d). A square i a strip is called terial if it cotais corers. A square tetacle T is a supergrid graph which is either a strip or a uio of a series of strips stuck together by the edges of terial squares. Let T = S (a 1, b 1 ; c 1, d 1 ) S (a, b ; c, d ) S (a k, b k ; c k, d k ). We defie T to satisfy the followig coditios: (1) both c i, d i V(S (a i+1, b i+1 ; c i+1, d i+1 )) for 1 i k 1, () oe of a i+1, b i+1 V(S (a i, b i ; c i, d i )) for 1 i k 1, ad () there is o other itersectio betwee the vertex sets of the strips. The vertices a 1, b 1, c k, d k are called the corers of T, ad deote this square tetacle by T (a 1, b 1 ; c k, d k ). For exaple, Fig. shows a square tetacle with strips ad corers a 1, b 1, c, d. I the costructio of a elarged supergrid graph G (see Fig. ) fro a plaar bipartite graph B (see Fig. ), the path i G is a cobiatio of four possible types of subpaths, as show i Fig.. The correspodig square tetacles for these types of subpaths are depicted i Fig.. A tetacle is the costructed fro a square tetacle by attachig a triagle to its terial square. For istace, Fig. (c) depicts the possible tetacles for the square tetacle of type I i Fig.. We call the attached triagle of a tetacle to be the tail of the tetacle. Let u, v be two vertices of the first square i a tetacle T such that their degrees are, ad let w be a vertex of the tail i a tetacle T such that its degree. Sice each strip of a tetacle is isoorphic to a -rectagle R(, ) with, the vertices u, v exist. The we deote the tetacle by T(u, v; w). The vertices u, v are called the twi corers of T(u, v; w), ad the vertex w is said to be the tail corer of T(u, v; w). Fig.(c) also depicts the corers of tetacles. The followig lea shows the Hailtoia property of a tetacle. Lea.. Let T(u, v; w) be a tetacle. The for ay two corers s, t {u, v, w}, there exists a Hailtoia (s, t)-path of T(u, v; w). Proof. Let T (a 1, b 1 ; c η, d η )=S (a 1, b 1 ; c 1, d 1 ) S (a, b ; c, d ) S (a η, b η ; c η, d η ) be the correspodig square tetacle of T(u, v; w). That is, T(u, v; w) is costructed fro T (a 1, b 1 ; c η, d η ) by attachig a triagle with vertices c η, d η, w. The, u, v {a 1, b 1 } ad w is adjacet to both of c η ad d η. We clai that T (a 1, b 1 ; c η, d η ) cotais a Hailtoia (s, t )-path for s, t {a 1, b 1 } such that edge (c η, d η ) is i the Hailtoia path, ad has a Hailtoia (s, t )-path for s {a 1, b 1 } ad t {c η, d η }. Sice u, v {a 1, b 1 } ad w is adjacet to both of c η ad d η, the lea hece holds true. We prove the above clai by iductio oη, the uber of strips i square tetacle T (a 1, b 1 ; c η, d η ). Iitially, letη=1. The, T (a 1, b 1 ; c 1, d 1 )=S (a 1, b 1 ; c 1, d 1 ). By ispectio, it is easy to verify that there exists a Hailtoia (a 1, b 1 )-path of strip S (a 1, b 1 ; c 1, d 1 ) such that edge (c 1, d 1 ) is i it. That is, the Hailtoia path cotais d a = c d c b a = d b

10 I II III I II III IV IV I I I I Fig. : The possible types of subpaths i a elarged supergrid graph G, the correspodig square tetacles for these subpaths i, ad (c) the possible tetacles for the square tetacle of type I i, where the solid lies idicate the edges i paths. a 1 b 1 a = c d 1 a k+1 r p 1 b b c k+1= k d k q c k+1 d k+1 a = c d c k 1. b ak= dk 1 b k a 1 b 1 a = c d 1 a k+1 r p 1 b b k+1 =c k d k q Q 1 c k+1 d k+1 P 1 1 P 1 a = c d c k 1. b ak= dk 1 b k (c) a 1 b 1 a = c d 1 a k+1 r p 1 b b k+1 =c k d k q Q c k+1 d k+1 P (c) u v a = c d c k 1. b ak= dk 1 Fig. : A scheatic diagra for the relative locatio of a k+1, b k+1, d k, r, p, q, c k+1, d k+1, the Hailtoia (a 1, b 1 )-path of T (a 1, b 1 ; c k+1, d k+1 ), ad (c) the Hailtoia (a 1, c k+1 )-path of T (a 1, b 1 ; c k+1, d k+1 ), where arrow lies idicate the edges i the Hailtoia paths. its all boudary edges except edge (a 1, b 1 ). I additio, for s {a 1, b 1 } ad t {c 1, d 1 } we ca easily costruct a Hailtoia (s, t )-path of S (a 1, b 1 ; c 1, d 1 ). Now, assue that the clai holds true wheη = k 1. The, there exists a Hailtoia (a 1, b 1 )-path P 1 of T (a 1, b 1 ; c k, d k ) such that edge (c k, d k ) is i P 1, ad there exists a Hailtoia (s, t )-path P of T (a 1, b 1 ; c k, d k ) for s {a 1, b 1 } ad t {c k, d k }. Cosider thatη=k+ 1. The, T (a 1, b 1 ; c k+1, d k+1 ) = T (a 1, b 1 ; c k, d k ) S (a k+1, b k+1 ; c k+1, d k+1 ). By defiitio of square tetacle, oe of a k+1, b k+1 {c k, d k } ad both c k, d k V(S (a k+1, b k+1 ; c k+1, d k+1 )). Without loss of geerality, suppose that b k+1 = c k ad the first square of S (a k+1, b k+1 ; c k+1, d k+1 ) cotais vertices a k+1, b k+1 = c k, d k, r. Let S = S (p, q; c τ, d τ ) be a strip obtaied fro S (a k+1, b k+1 ; c k+1, d k+1 ) by reovig the vertices of the first square. Sice each strip cotais at least two squares, we have that S, c τ = c k+1, ad d τ = d k+1. Note that if S (a k+1, b k+1 ; c k+1, d k+1 ) cotais oly two squares, the p=c k+1 ad q=d k+1. The relative locatio of a k+1, b k+1, d k, r, p, q, c k+1, d k+1 is depicted i Fig.. The, d k, r, p, q fors the secod square of S (a k+1, b k+1 ; c k+1, d k+1 ). Note that each square of a strip is a clique. Cosider the followig two cases: Case 1: s, t {a 1, b 1 }. By the iductio hypothesis, there exists a Hailtoia (a 1, b 1 )-path P 1 of T (a 1, b 1 ; c k, d k ) such that edge (c k, d k ) is i P 1. Without loss of geerality, suppose that P 1 = P 1 1 c k d k P 1, i.e., c k appears before d k i P 1, where start(p 1 1 ), ed(p 1 ) {a 1, b 1 }. The case that c k appears after d k i P 1 ca be verified siilarly. By ispectio, we ca costruct a Hailtoia (p, q)-path Q 1 of S = S (p, q; c k+1, d k+1 ) such that edge (c k+1, d k+1 ) Q 1. Let P = P 1 1 c k a k+1 r Q 1 d k P 1. The, P is a Hailtoia (a 1, b 1 )-path of T (a 1, b 1 ; c k+1, d k+1 ) such that edge (c k+1, d k+1 ) is i P. The costructio of such a Hailtoia (a 1, b 1 )-path P is depicted i Fig.. Case : s {a 1, b 1 } ad t {c k+1, d k+1 }. By the iductio hypothesis, there exists a Hailtoia (s, d k )-path P b k w

11 e e C C 1 x e 1 e u v T( u, v; w) Fig. 1: The coectio of two clusters via a tetacle, where dashed lies represet the edges betwee cluster ad tetacle. of T (a 1, b 1 ; c k, d k ). By ispectio, we ca costruct a Hailtoia (p, t )-path Q of S = S (p, q; c k+1, d k+1 ), where t {c k+1, d k+1 }. Let P = P a k+1 r Q. The, P is a Hailtoia (s, t )-path of T (a 1, b 1 ; c k+1, d k+1 ), where s {a 1, b 1 } ad t {c k+1, d k+1 }. The costructio of such a Hailtoia (a 1, c k+1 )-path P is show i Fig. (c). It iediately follows fro the above cases that the clai holds true. This copletes the proof of the lea. Let B=(V 0 V 1, E) be a plaar bipartite graph with vertices ad axiu degree, G 1 be the ebeddig supergrid graph fro B, ad let G be the elarged supergrid graph by ultiplyig the scale of G 1 by. We have siulated the critical vertices of G by clusters ad the paths of G by tetacles. The reaiig care is take as to how the tetacle is coected to the clusters correspodig to two critical vertices of G. Let x V 0 ad y V 1 with (x, y) E(B). The, we siulate x ad y by a white cluster C ad a black cluster C 1, respectively, such that C cotais the white vertex x as ceter ad four critical edges e 1, e, e, e, ad C 1 cotais the black vertex x as ceter ad four corer vertices b 1, b, b, b, as show i Fig.. The path betwee x ad y i G is the siulated by a tetacle T(u, v; w). Tetacle T(u, v; w) is used to coect clusters C ad C 1 i the followig way. The twi corers, u ad v, of T(u, v; w) are adjacet to the vertices of oe critical edge i C, ad the other corer, w, of T(u, v; w) is adjacet to oe corer vertex of C 1. For exaple, Fig. 1 depicts such a coectio betwee white cluster C ad black cluster C 1 via a tetacle T(u, v; w). Sice the axiu degree of the origial plaar bipartite graph is, the uber of tetacles coectig to each cluster is at ost. For a white cluster (resp., black cluster), there are four critical edges (resp., corer vertices) (see Fig. ) ad the uber of critical edges (resp., corer vertices) used to coect tetacles is at ost sice the axiu degree of the origial plaar bipartite graph is. Thus, it is eough to ake such a coectio. O the other had, the paths i ebeddig supergrid graph G 1 are vertex disjoit ad hece they are vertex disjoit paths i the elarged supergrid graph G. Sice we elarge the scale of G 1 by, i.e., each edge i G 1 is trasfored ito a path with edges, it is easy to costruct tetacles fro the paths of G such that tetacles are disjoit. Let T(u, v; w) be a tetacle with twi corers u, v ad tail corer w. By Lea., there exists a Hailtoia (s, t)-path of T(u, v; w) for s, t {u, v, w}. By the defiitio of tetacle, u, v are adjacet ad w is adjacet to either u or v. We ca easily observe that there are oly two types of Hailtoia (s, t)-paths i T(u, v; w). The path ca be either a retur path if s, t are twi corers or a cross path if oe of s, t is tail corer. For exaple, Fig. 1 depicts these two types of Hailtoia paths i T(u, v; w) show i Fig. 1. Note that there are ay retur paths ad cross paths. We have itroduced how to costruct a supergrid graph G s fro a plaar bipartite graph B with axiu degree. The costructio algorith is forally preseted as follows. Algorith SupergridCostructio Iput: B=(V 0 V 1, E), a plaar bipartite graph with axiu degree. Output: G s, a supergrid graph costructed fro B. Method: 1. ebed graph B ito a rectagular supergrid graph R(k, k) for soe costat k [], ad let the ebeddig supergrid graph be G 1 ;. elarge G 1 to a supergrid graph G such that each edge i G 1 is trasfored ito a path with edges;. for each white critical vertex x of G (x V 0 ), x is trasfored ito a white cluster C with ceter x;. for each black critical vertex y of G (y V 1 ), y is trasfored ito a black cluster C 1 with ceter y; w b b 1 y b b

12 e e e e x x e 1 e 1 e e u v u v T( u, v; w) T( u, v; w) Fig. 1: A retur path, ad a cross path i a tetacle T(u, v; w) show i Fig. 1, where arrow lies idicate the paths.. for each path betwee white critical vertex x ad black critical vertex y, costruct a tetacle T(u, v; w) to coect the correspodig clusters of x ad y such that u, v are adjacet to two vertices of oe critical edges i white cluster ad w is adjacet to oe corer vertex of black cluster;. let the costructed supergrid graph be G s ad output G s. For exaple, give a plaar bipartite graph B=(V 0 V 1, E) with axiu degree show i Fig., the ebeddig supergrid graph G 1 is show i Fig.. The elarged supergrid graph G by ultiplyig the scale of G 1 by is show i Fig.. The costructed supergrid graph G s is depicted i Fig. 1. By Lea., lie 1 of Algorith SupergridCostructio ca be doe i polyoial tie. Clearly, lies of the algorith ca be doe i polyoial tie. The, Algorith SupergridCostructio rus i polyoial tie ad hece the followig lea holds true. Lea.. Give a plaar bipartite graph B=(V 0 V 1, E) with axiu degree, Algorith SupergridCostructio costructs a supergrid graph G s i polyoial tie. Next, we will prove that supergrid graph G s has a Hailtoia cycle if ad oly if there exists a Hailtoia cycle i the plaar bipartite graph B. Before provig the above property, we first give the relatio betwee tetacle ad Hailtoia cycle of G s. For a cycle C of a graph G ad a subgraph H of G, we deote the restrictio of C to H by C H. The, C H is a set of subpaths of C. We the have the followig lea. Lea.. Let G s be the supergrid graph costructed fro a plaar bipartite graph B by Algorith Supergrid- Costructio, ad let T = T(u, v; w) be a tetacle of G s. If G s has a Hailtoia cycle HC, the there exists a Hailtoia cycle HC i G s such that HC T is a Hailtoia (s, t)-path of T for s, t {u, v, w}. Proof. By the costructio of G s, tetacle T satisfies the followig properties: (1) oly three vertices u, v, w of T are adjacet to vertices of G s T, i.e., o vertex of T {u, v, w} is adjacet to vertices of G s T, () each of twi corers u, v of T is adjacet to oly two vertices p, q of G s T, i.e., N(u) (G s T)= N(v) (G s T)= {p, q} ad u, v, p, q fors a clique, ad () the tail corer w of T is adjacet to oly oe vertex r of G s T, i.e., N(w) (G s T)={r}. Sice oly three vertices u, v, w of T are adjacet to vertices of G s T, G s T ad T {u, v, w} are disjoit. The, the uber of paths i HC T is ot larger tha, i.e., HC T. If HC T =1, i.e., HC T is either a retur path or a cross path of T, the the lea is clearly true. Assue that HC T = below. Let u 1, u, u {u, v, w} ad let HC=P 1 u 1 Q 1 u P u Q, where HC T ={u 1 Q 1 u, u Q } ad P. Suppose that P 1 =. Sice HC is a Hailtoia cycle of G s, u 1 ed(q ) ad hece HC T ={u rev(q 1 ) 1 w w b b b 1 y b b 1 y b b b

13 x 1 y 1 x x Fig. 1: A supergrid graph G s costructed fro a plaar bipartite graph B=(V 0 V 1, E) with axiu degree show i Fig., where solid lies idicate the edges i the costructed supergrid graph. 1 y y

14 u 1 rev(q ) u }, a cotradictio. Thus, P 1. O the other had, suppose that Q. Sice HC is a cycle, ed(q ) start(p 1 ) ad hece there exist four vertices, u 1, u, u, ed(q ), of T which are adjacet to vertices of G s T, a cotradictio. Thus, Q =. The, HC= P 1 u 1 Q 1 u P u, where P 1, start(p 1 ) u, ad HC T ={u 1 Q 1 u, u }. Sice u start(p 1 ) ad u ed(p ), N(u ) (G s T) ad hece u {u, v}. The, start(p 1 ), ed(p ) {p, q} ad start(p 1 ) ed(p ). Let P=P P 1. The, start(p)(= start(p )) u ad ed(p)(= ed(p 1 )) u 1. By Lea., there exists a Hailtoia (u 1, u )-path Q of T for u 1, u {u, v, w}. Let HC = P Q. The, HC is the desired Hailtoia cycle of G s such that HC T ={Q} ad Q is a Hailtoia (s, t)-path of T for s=u 1 ad t=u. I fact, Q is a cross path of T. Thus, the lea holds true. By usig the above lea, we will prove the followig lea. Lea.. Let B=(V 0 V 1, E) be a plaar bipartite graph with axiu degree ad let G s be the supergrid graph costructed fro B by Algorith SupergridCostructio. The, graph G s has a Hailtoia cycle if ad oly if there exists a Hailtoia cycle i graph B. Proof. If part: I this part, we will prove that if graph B has a Hailtoia cycle, the graph G s cotais a Hailtoia cycle. Assue that the plaar bipartite graph B=(V 0 V 1, E) has a Hailtoia cycle HC B. We will costruct the correspodig Hailtoia cycle HC of G s as follows. Let (x, y) be a edge of graph B such that x V 0 ad y V 1, ad let the edge be siulated by a tetacle T xy = T(u, v; w) i G s. Startig to for HC, we will cover T xy by a cross path if (x, y) HC B, ad by a retur path otherwise. By Lea., there exists a cross path or retur path of tetacle T xy. The clusters theselves are covered as i Propositios. ad.. The partial paths ca be coected to costitute a Hailtoia cycle. Note that soe critical edges of e i s for coectig to a retur path i Fig. ust be deleted i the costructed Hailtoia cycle. For exaple, for the plaar bipartite graph B=(V 0 V 1, E) show i Fig., HC B = x 1 y 1 x y x y is a Hailtoia cycle of B. The, Fig. 1 depicts its correspodig Hailtoia cycle HC i the costructed supergrid graph G s as show i Fig. 1. Oly If part: I this part, we will prove that if graph G s has a Hailtoia cycle, the graph B cotais a Hailtoia cycle. Assue ow that supergrid graph G s has a Hailtoia cycle HC. By Lea., we ay assue that for ay tetacle T= T(u, v; w) i G s, the restrictio HC T of HC to T is a Hailtoia (s, t)-path for s, t {u, v, w}. The, by our costructio of G s each tetacle is covered by either a cross path or a retur path. I our costructio of G s, each tetacle is coected to oly oe critical edge of a white cluster, ad each white cluster is attached to at ost three tetacles. O the other had, each tetacle is coected to oly oe corer vertex of a black cluster, ad each black cluster is coected to at ost three tetacles. We ca see that i HC each white or black cluster is icidet upo exactly two cross paths. Note that each vertex of B is trasfored ito a cluster, ad every edge of B is siulated by a tetacle. To costruct a Hailtoia cycle HC B of graph B, we iclude i HC B all edges correspodig to tetacles covered by cross paths. Ad each vertex of HC B is a ceter of oe cluster i G s. The, HC B is a Hailtoia cycle of graph B because each cluster (white or black) of G s ca ot be covered by HC uless it is icidet upo exactly two cross paths. Clearly, the Hailtoia cycle proble for supergrid graphs is i NP. By Leas. ad., we coclude the followig theore. Theore.. The Hailtoia cycle proble for supergrid graphs is NP-coplete. By siilar arguets i provig the above theore, we ca prove the Hailtoia path proble o supergrid graphs to be also NP-coplete, as i the followig theore. Theore.. The Hailtoia path proble for supergrid graphs is NP-coplete. Proof. We give a reductio fro the Hailtoia cycle proble o plaar bipartite graphs with axiu degree. Give a plaar bipartite graph B with axiu degree, we costruct a supergrid graph G s as follows: 1. costruct a supergrid graph G s by Algorith SupergridCostructio;. let C b be a black cluster of G s such that its oe critical edge (s, t ) is ot attached to ay tetacle;. two ew vertices s ad t are added to be adjacet to s ad t, respectively, such that the degree of each ew vertex is 1; 1

15 x 1 (x 1, y1) HCB a cross path ( x, y ) HC 1 B a retur path y 1 x Fig. 1: A Hailtoia cycle HC of a supergrid graph G s costructed fro a Hailtoia cycle HC B = x 1 y 1 x y x y of a plaar bipartite graph B show i Fig., where solid lies idicate the edges i the cycle ad represets the destructio of a edge while costructig such a cycle. 1 x y y

16 let the resultat supergrid graph be G s. t t' s s' attached tetacles critical edge ot attached by ay tetacle Fig. 1: A black cluster C b i G s with two terials s ad t. Sice the degree of each vertex i B is at ost, each tetacle is coected to oe critical edges, ad there are four critical edges i each black cluster, black cluster C b of G s does exist. The costructio for attachig two ew vertices is depicted i Fig. 1. By siilar arguets i provig Lea., we ca verify that graph G s has a Hailtoia (s, t)-path if ad oly if there exists a Hailtoia cycle i graph B. The, Lea. ad NP-copleteess of the Hailtoia cycle proble for plaar bipartite graphs with axiu degree coplete the proof of this theore.. The Hailtoia cycle proble o rectagular ad alphabet supergrid graphs I this sectio, we will study the Hailtoia cycle property of rectagular ad alphabet supergrid graphs. We show that these two subclasses of supergrid graphs always cotai Hailtoia cycles. I the literature, Che et al. [] ad Sala [] showed the Hailtoia properties of rectagular ad alphabet grid graphs, respectively, as show i the followig two leas. Lea.1. (See [].) Let R (, ) with, be a rectagular grid graph which is a subgraph of R(, ), where V(R (, ))={v=(v x, v y ) 1 v x ad 1 v y }. The, R (, ) cotais a Hailtoia cycle if ad oly if is eve. Lea.. (See [].) Let L (, ) with, be a L-aphabet grid graph which is a subgraph of L(, ), where L (, ) is defied siilar to L(, ). The, L (, ) has a Hailtoia cycle if ad oly if is eve. Sice ay rectagular grid graph R (, ) is a subgraph of a rectagular supergrid graph R(, ), R(, ) cotais a Hailtoia cycle if is eve. However, we will show that R(, ) cotais a Hailtoia cycle eve if is odd. Obviously, 1-rectagle cotais o Hailtoia cycle. Without loss of geerality, assue that for R(, ). For a -rectagle R(, ), we ca costruct a Hailtoia cycle by visitig all boudary edges of R(, ). I the followig, cosider R(, ) to satisfy that. By defiitio, R(, ) cosists of colus ad rows of vertices. Let a i j be the vertex locatig at i-th row ad j-th colu of R(, ). That is, (i, j) is the coordiates of a i j whe a is coordiated as (1, 1). Let C be a cycle of R(, ) ad let H be a boudary of R(, ), where H is a subgraph of R(, ). Recall that the restrictio of C to H is deoted by C H. If C H =1, i.e. C H visits all boudary edges of H, the C H is called flat face o H. If C H >1 ad C H cotais at least oe boudary edge of H, the C H is called cocave face o H. We first cosider -rectagle R(, ). A Hailtoia cycle of R(, ) is called caoical if it cotais three flat faces o two shorter boudaries ad oe loger boudary, ad it cotais oe cocave face o the other boudary, 1

17 a 1 a 1 a ( 1)1 a1 a a 1 a 1 a 1 a 1 a 1( 1) a a a ( 1) a a a ( 1) a 1 a a a a 1 a 1 a 1 a 1 a 1( 1) a ( 1) a a flat face a ( 1) a cocave face a 1 a a a 1 a a ( )1 a ( ) a ( ) a ( ) a ( )1 a ( ) a ( ) a ( ) a ( 1)1 a ( 1) a ( 1) a 1 a a a Fig. : The caoical Hailtoia cycle of rectagular supergrid graph R(, ) for is eve, ad is odd, where, solid arrow lies idicate the edges i the cycle, ad dashed arrow lies idicate the flat faces i the cycle. where the shorter boudary cotais three vertices. The followig lea shows that R(, ) cotais a caoical Hailtoia cycle. Lea.. Let R(, ) be a -rectagle with. The, R(, ) cotais a caoical Hailtoia cycle. Proof. We prove this lea by costructig a Hailtoia cycle of R(, ) such that it cotais all boudary edges of two shorter boudaries ad oe loger boudary, ad it cotais at least oe boudary edge i the other loger boudary. By ispectio, the lea ca be easily verified for. I the followig, assue that. Let P l1 = a a 1 a 1( 1) a 1, ad let P ι = a ι a ι for ι 1. Depedig o whether is eve or ot, we cosider the followig two cases: Case 1: is eve. Let P l = rev(p ) P rev(p j ) P j+1 rev(p ) P 1, where j is eve ad j. The, P l1 P rev(p l ) rev(p 1 ) is a caoical Hailtoia cycle of R(, ). Case : is odd. Let P l = rev(p ) P rev(p j ) P j+1 rev(p ) P rev(p 1 ), where j is eve ad j. The, P l1 P rev(p l ) rev(p 1 ) is a caoical Hailtoia cycle of R(, ). It iediately follows fro the above cases that the lea holds true. We have costructed Hailtoia cycles for -rectagles ad -rectagles. I the followig, let R(, ) satisfy. A Hailtoia cycle of R(, ) with is called caoical if it cotais three flat faces o three boudaries, ad it cotais oe cocave face o the other boudary. The followig lea shows the Hailtoia property of R(, ) with. Lea.. Let R(, ) be a rectagular supergrid graph with. The, R(, ) cotais four caoical Hailtoia cycles with cocave faces beig o differet boudaries. Proof. Depedig o whether is eve or ot, we cosider the followig two cases to costruct a caoical Hailtoia cycle of R(, ): Case 1: is eve. Let P 1 = a a 1 a 1, P ι = a ι a ι a ι for ι, ad let P +1 = a 1 a 1 a ( 1)1 a 1. Let P = rev(p ) P rev(p ) P P j rev(p j+1 ) rev(p ) P 1, where j is eve ad j. Let C eve = P 1 P rev(p ) rev(p +1 ). The, C eve is a caoical Hailtoia cycle of R(, ). The costructio of such a caoical Hailtoia cycle is depicted i Fig.. Case : is odd. I this case,. Let P 1 = a a 1 a 1, P ι = a ι a ι a ι for ι, P = a a a, ad let P +1 = a 1 a 1 a ( 1)1 a 1. Let P = rev(p ) P rev(p ) P rev(p j ) P j+1 rev(p ) P rev(p ), where j is

18 Fig. : The Hailtoia cycle for parallelis supergrid graph P(, ), ad parallelis supergrid graph P(, ), where arrow lies idicate the edges i such cycles. eve ad j. If is eve, the let P = a ( ) a ( 1) a ( 1) a ( ) a ( )τ1 a ( 1)τ1 a ( 1)(τ1 +1) a ( )(τ1 +1) a ( )( ) a ( 1)( ) a ( 1)( 1) a ( )( 1) a ( ) a ( 1) ; otherwise, let P = a ( ) a ( 1) a ( 1) a ( ) a ( )τ a ( 1)τ a ( 1)(τ +1) a ( )(τ +1) a ( )( ) a ( 1)( ) a ( 1)( ) a ( )( ) a ( )( 1) a ( 1)( 1) a ( ) a ( 1), where τ 1,τ are eve, τ 1, ad τ. Let C odd = P 1 P P rev(p ) rev(p +1 ). The, C odd is a caoical Hailtoia cycle of R(, ). The costructio of such a caoical Hailtoia cycle is depicted i Fig.. By the above cases, we costruct a caoical Hailtoia cycle of R(, ) such that its cocave face is o the right boudary. By syetry, we ca costruct a caoical Hailtoia cycle of R(, ) such that its cocave face is o the left boudary. Cosider that is eve or ot. By syetry ad the sae costructios i Case 1 ad Case, we ca costruct two caoical Hailtoia cycles of R(, ) such that their cocave faces are respectively placed at the upper ad dow boudaries. Thus there are four caoical Hailtoia cycles of R(, ) such that their cocave faces are o the differet boudaries (left, right, upper, ad dow boudaries). This copletes the proof of the lea. By siilar costructios i the proofs of Leas. ad., we ca costruct a Hailtoia cycle of a parallelis supergrid graph. For istace, Fig. shows the Hailtoia cycles of parallelis supergrid graphs P(, ) ad P(, ). Next, we will ivestigate the Hailtoia cycle property of alphabet supergrid graphs. By Lea., a L- alphabet grid graph L (, ) has a Hailtoia cycle oly if is eve. However, for a L-alphabet supergrid graph L(, ) we will show that it always cotais a Hailtoia cycle. Two distict edges e 1 = (u 1, v 1 ) ad e = (u, v ) of a graph G are called parallel if (u 1 v 1 ad u v ) or (u 1 v ad u v 1 ), deote this by e 1 e. Let C 1 ad C be two vertex-disjoit cycles of a graph G. If there exist two edges e 1 C 1 ad e C such that e 1 e, the C 1 ad C ca be cobied ito a cycle of G. Thus we have the followig propositio. Propositio.. Let C 1 ad C be two vertex-disjoit cycles i graph G. If there exist two edges e 1 C 1 ad e C such that e 1 e, the C 1 ad C ca be cobied ito a cycle C. To costruct a Hailtoia cycle of a L-alphabet supergrid graph L(, ), we partitio it ito two adjacet rectagular supergrid subgraphs. Note that L(, ) is a subgraph of R(, ) for,. The L-alphabet supergrid graph L(, ) is separated ito two disjoit rectagular supergrid subgraphs L 1 ad L such that L 1 = R(, ) ad L = R(, ). The partitio is depicted i Fig.. Sice,, we obtai that ad. By Leas. ad., L 1 ad L cotai caoical Hailtoia cycles C 1 ad C, respectively. We ca place oe flat face of C 1 to face the eighborig rectagular supergrid subgraph L ad place oe flat face of C to face L 1. Thus, there exist two parallel boudary edges e 1 C 1 ad e C. By Propositio., C 1 ad C ca be cobied ito a Hailtoia cycle of L(, ). For exaple, Fig. shows a Hailtoia cycle of L(, ). We the have the followig lea. Lea.. Let L(, ) be a L-alphabet supergrid graph with,. The, L(, ) cotais a Hailtoia cycle. By siilar partitio, we ca separate the other types of alphabet supergrid graphs below.

19 L1 = R(, ) L = R(, ) C 1 L1 = R(, ) L = R(, ) parallel boudary edges Fig. : The partitio of L-alphabet supergrid graph L(, ), where dashed lie idicates the separatio, ad a Hailtoia cycle of L(, ), where arrow lies idicate the edges i the cycle ad represets the destructio of a edge while costructig such a Hailtoia cycle. C1 = L(, ) C = R(, ) F1 = L(, ) E1 = F(, ) F = R(, ) C E = R(, ) Fig. 0: The partitios of alphabet supergrid graphs for C-alphabet, F-alphabet, ad (c) E-alphabet, where dashed lies idicate the separatios. Defiitio.1. The partitios of C-, F-, ad E-alphabet supergrid graphs are defied as follows: (1) A partitio of a C-alphabet supergrid graph C(, ) is a separatio of C(, ) ito a L-alphabet supergrid subgraph C 1 = L(, ) ad a rectagular supergrid subgraph C = R(, ). () A partitio of a F-alphabet supergrid graph F(, ) is a separatio of F(, ) ito a L-alphabet supergrid subgraph F 1 = L(, ) ad a rectagular supergrid subgraph F = R(, ). () A partitio of a E-alphabet supergrid graph E(, ) is a separatio of E(, ) ito a F-alphabet supergrid subgraph E 1 = F(, ) ad a rectagular supergrid subgraph E = R(, ). Fig. 0 depicts the partitios of the above alphabet supergrid graphs. By siilar arguets i provig Lea., the followig lea ca be easily verified. Lea.. Let C(, ), F(, ), ad E(, ) be a C-alphabet, a F-alphabet, ad a E-alphabet supergrid graph, respectively, for,. The, C(, ), F(, ), ad E(, ) cotai Hailtoia cycles. We have proved that rectagular, parallelis, ad alphabet supergrid graphs are Hailtoia. The followig theore cocludes these results. Theore.. Let A(, ) be a rectagular or parallelis supergrid graph with,, ad let B(, ) be a L- alphabet, C-alphabet, F-alphabet, or E-alphabet supergrid graph with,. The, A(, ) ad B(, ) cotai Hailtoia cycles. (c)

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