Hamiltonian Paths in Some Classes of Grid Graphs

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Volume 2012, Article ID 475087,17pages doi:10.1155/2012/475087

Research Article

Hamiltonian Paths in Some Classes of Grid Graphs

Fatemeh Keshavarz-Kohjerdi

¹

and Alireza Bagheri

²

1Department of Computer Engineering, Islamic Azad University, North Tehran Branch, Tehran, Iran

2Department of Computer Engineering and IT, Amirkabir University of Technology, Hafez Avenue, Tehran, Iran

Correspondence should be addressed to Fatemeh Keshavarz-Kohjerdi, [email protected]

Received 8 September 2011; Revised 13 December 2011; Accepted 28 December 2011 Academic Editor: Jin L. Kuang

Copyrightq2012 F. Keshavarz-Kohjerdi and A. Bagheri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and suﬃcient conditions for the existence of Hamiltonian paths inL-alphabet, C-alphabet,F-alphabet, andE-alphabet grid graphs. We also present linear-time algorithms for ﬁnding Hamiltonian paths in these graphs.

1. Introduction

Hamiltonian path in a graph is a simple path that visits every vertex exactly once. The problem of deciding whether a given graph has a Hamiltonian path is a well-known NP-complete problem and has many applications1,2. However, for some special classes of graphs polynomial-time algorithms have been found. For more related results on Hamiltonian paths in general graphs see3–8.

Rectangular grid graphs first appeared in9, where Luccio and Mugnia tried to solve the Hamiltonian path problem. Itai et al.10gave necessary and sufficient conditions for the existence of Hamiltonian paths in rectangular grid graphs and proved that the problem for general grid graphs is NP-complete. Also, the authors in11presented sufficient conditions for a grid graph to be Hamiltonian and proved that all finite grid graphs of positive width have Hamiltonian line graphs. Later, Chen et al.12improved the algorithm of10 and presented a parallel algorithm for the problem in mesh architecture. Also there is a polynomial-time algorithm for finding Hamiltonian cycle in solid grid graphs13. Recently, Salman14introduced alphabet grid graphs and determined classes of alphabet grid graphs which contain Hamiltonian cycles. More recently, Islam et al.15showed that the Hamiltoni- an cycle problem in hexagonal grid graphs is NP-complete. Also, Gordon et al.16proved that all connected, locally connected triangular grid graphs are Hamiltonian, and gave a

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suﬃcient condition for a connected graph to be fully cycle extendable and also showed that the Hamiltonian cycle problem for triangular grid graphs is NP-complete. Nandi et al.

17 gave methods to ﬁnd the domination numbers of cylindrical grid graphs. Moreover, Keshavarz-Kohjerdi et al.18, 19 gave sequential and parallel algorithms for the longest path problem in rectangular grid graphs.

In this paper, we obtain necessary and sufficient conditions for the existence of a Ham- iltonian path inL-alphabet, C-alphabet,F-alphabet, andE-alphabet grid graphs. Also, we present linear-time algorithms for finding such a Hamiltonian path in these graphs. Solving the Hamiltonian path problem for alphabet grid graphs may arise results that can help in solving the problem for general solid grid graphs. The alphabet grid graphs that are considered in this paper have similar properties that motivate us to investigate them together. Other classes of alphabet grid graphs have enough differences that will be studied in a separate work.

2. Preliminaries

Some previously established results about the Hamiltonian path problem which plays an im- portant role in this paper are summarized in this section.

Thetwo-dimensional integer gridG^∞is an inﬁnite graph with vertex set of all points of the Euclidean plane with integer coordinates. In this graph, there is an edge between any two vertices of unit distance. For a vertexvof this graph, letvx andvy denotexand ycoordi- nates of its corresponding point. A grid graphG is a ﬁnite vertex-induced subgraph of the two-dimensional integer grid. In a grid graphG, each vertex has degree of at most four. A rectangular grid graphRm, n orRfor shortis a grid graph whose vertex set isVR {υ| 1≤υx ≤m, 1≤υy ≤n}.Rm, nis called ann-rectangle. A solid grid graph is a grid graph without holes.

By10, a vertexvis coloredwhiteifυ_x υ_yis even, and is coloredblackotherwise. The size ofRm, nis defined to bemn.Rm, nis calledodd sizedifmnis odd, and is calledeven sizedotherwise. Two different verticesυandυ^′inRm, nare calledcolor compatibleif either bothυandυ^′are white andRm, nis odd sized, orυandυ^′have different colors andRm, n is even sized.

An alphabet grid graph is a ﬁnite vertex-induced subgraph of the rectangular grid graph of a certain type, as follows. Form, n ≥3, anL-alphabet grid graphLm, n orLfor short,C-alphabet grid graphCm, n orCfor short,F-alphabet grid graphFm, n orFfor short, andE-alphabet grid graphEm, n orEfor shortare subgraphs ofR3m−2,5n−4.

These alphabet grid graphs are shown inFigure 1, form 4 andn 3. An alphabet grid graph is calledodd sizedif its corresponding rectangular graph is odd sized, and is calledeven sizedotherwise.

In the following byAm, nwe mean a grid graphAm, n. LetPAm, n, s, tdenote the problem of finding a Hamiltonian path between verticessandtin grid graphAm, n, and letAm, n, s, tdenote the grid graphAm, nwith two specified distinct verticessand tof it. WhereAis a rectangular grid graph,L-alphabet,C-alphabet,F-alphabet, orE-alphabet grid graph. Am, n, s, tis Hamiltonianif there is a Hamiltonian path between s andt in Am, n. In this paper, without loss of generality we assumesx≤txandm≥n. In the figures, we assume that1,1is the coordinates of the vertex in the lower left corner.

An even-sized rectangular grid graph contains the same number of black and white vertices. Hence, the two end vertices of any Hamiltonian path in the graph must have

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3m−2

5n−4

a

4n−4

2m−2

m

n

b

m

n n

3n−4

2m−2

c

n−2

m

2m−4

n

2n−2

d

m

n−2

2m−4

n

2m−2 e

Figure 1:aA rectangular grid graphR10,11,banL-alphabet grid graphL4,3,canC-alphabet grid graphC4,3,danF-alphabet grid graphF4,3,eanE-alphabet grid graphE4,3.

s t

a

s t

b

s t c

s t

d

Figure 2:Rectangular grid graphs in which there is no Hamiltonian path betweensandt.

diﬀerent colors. Similarly, in an odd sized rectangular grid graph the number of white vertices is one more than the number of black vertices. Therefore, the two end vertices of any Ham- iltonian path in such a graph must be white. Hence, the color compatibility ofsandtis a necessary condition forRm, n, s, tto be Hamiltonian. Furthermore, Itai et al.10showed that if one of the following conditions hold, thenRm, n, s, tis not Hamiltonian:

F1Rm, nis a 1-rectangle and eithersortis not a corner vertexFigure 2a.

F2Rm, nis a 2-rectangle ands, tis a nonboundary edge, thats, tis an edge and it is not on the outer faceFigure 2b.

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a b

Figure 3:A Hamiltonian cycle for the rectangular grid graphR8,5andL-alphabet grid graphL4,3.

F3Rm, nis isomorphic to a 3-rectangleR^′m, nsuch thatsandtare mapped tos^′ andt^′and all of the following three conditions hold:

1mis even,

2s^′is black,t^′is white,

3s^′_y 2 ands^′_x< t^′_xFigure 2cors^′_y/2 ands^′_x< t^′_x−1Figure 2d.

A Hamiltonian path problemPRm, n, s, t is calledacceptableif sand tare color compatible andR, s, tdoes not satisfy any of conditionsF1,F2, andF3.

The following theorem has been proved in10.

Theorem 2.1. LetRm, nbe a rectangular grid graph and sandt be two distinct vertices. Then Rm, n, s, tis Hamiltonian if and only ifPRm, n, s, tis acceptable.

Lemma 2.2see12. Rm, nhas a Hamiltonian cycle if and only if it is even-sized andm, n >1.

Lemma 2.3see14. AnyL-alphabet grid graphLm, n m, n≥ 3has a Hamiltonian cycle if and only ifmnis even.

Figure 3shows a Hamiltonian cycle for an even-sized rectangular grid graph andL- alphabet graphLm, n, found by Lemmas2.2and2.3, respectively. Each Hamiltonian cycle found by these lemmas contains all boundary edges on three sides of the rectangular graph and four sides of theL-alphabet grid graph. This shows that for an even-sized rectangular graphRandL-alphabet graphLm, n, we can always ﬁnd a Hamiltonian cycle, such that it contains all boundary edges, except of exactly one side ofRand two side ofLwhich contains an even number of vertices.

3. Necessary and Sufficient Conditions

In this section, we give necessary and suﬃcient conditions for the existence of a Hamiltonian path inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs.

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s

t L(m, n) R−L

e1 e2

a

s

t

b Figure 4:A Hamiltonian path inR10,11.

Deﬁnition 3.1. Aseparationof

ian L-alphabet grid graphLm, nis a partition ofL into two disjoint rectangular grid graphsR₁andR₂, that is,VL VR₁∪VR₂, andVR₁∩VR₂ ∅, iianC-alphabet graphCm, nis a partition ofCinto aL-alphabet graphLm, nand

a rectangular grid graphR2m−2, n, that is,VC VL∪VR2m−2, n, and VL∩VR2m−2, n ∅,

iiianF-alphabet grid graphFm, nis a partition ofF into aL-alphabet grid graph Lm, nand a rectangular grid graphR2m−4, n or four rectangular grid graphs R1toR4, that is,VF VL∪VR2m−4, nandVL∩VR2m−4, n ∅or VF VR₁∪VR₂∪VR₃∪VR₄andVR₁∩VR₂∩VR₃∩VR₄ ∅, ivanE-alphabet grid graphEm, nis a partition ofEinto anF-alphabet grid graph

Fm, n and a rectangular grid graph R2m −2, n or a C-alphabet grid graph Cm, nand a rectangular grid graphR2m−4, n, that is,VE VF∪VR2m− 2, n, andVF∩VR2m−2, n ∅ or VE VC∪VR2m−4, n, and VC∩VR2m−4, n ∅.

In the following, two nonincident edgese1ande2are parallel, if each end vertex ofe1

is adjacent to some end vertex ofe₂.

Lemma 3.2. LetAm, nbe anL-alphabet,C-alphabet,F-alphabet, orE-alphabet grid graph andR be the smallest rectangular grid graph that includesA. IfAm, n, s, tis Hamiltonian, thenR, s, t is also Hamiltonian.

Proof. We break the proof into two cases.

Case 1Am, nis anL-alphabet orC-alphabet grid graph. LetPbe a Hamiltonian path inL orCthat is found byAlgorithm 1or2. SinceR−LorR−Cis an even-sized rectangular grid graph of2m−2×4n−4 or2m−2×3n−4, then byLemma 2.2it has a Hamiltonian cyclei.e., we can ﬁnd a Hamiltonian cycle ofR−LorR−C, such that it contains all edges ofR−LorR−Cthat are parallel to some edge ofP. Using two parallel edges ofPand the Hamiltonian cycle ofR−L orR−Csuch as two darkened edges of Figure 4a, we can combine them as illustrated inFigure 4band obtain a Hamiltonian path forR.

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e1

e2

v1 v2

a

s t

b

s t

c Figure 5:A Hamiltonian path inR7,11.

Case 2. Am, nis anF-alphabet orE-alphabet grid graph. LetP be a Hamiltonian path inF orEthat is found byAlgorithm 3or4. We consider the following cases.

Subcase 2.1m, n >3. SinceR−ForR−Ecan be partitioned into three even-sized rectangular grid graphs ofR2m−4, n−2,R2,2n−2, andR2m−2,2n−2 orR2,3n−4, R2m−4, n−

2, andR2m−4, n−2, then they have Hamiltonian cycles byLemma 2.2. Then combine Hamiltonian cycles onR2m−4, n−2,R2,2n−2andR2m−2,2n−2 orR2,3n−4, R2m−

4, n−2, andR2m−4, n−2to be a large Hamiltonian cycle and then using two parallel edges ofPand the Hamiltonian cycle ofR−ForR−E, we can obtain a Hamiltonian path forR.

Subcase 2.2n3. LetR−Fbe three rectangular grid graphs ofR2m−4, n−2,R2,2n−2, andR2m−2,2n−2. We consider the following two subcases.

Subcase2.2.1 m3. Let two verticesv1, v2be inR2m−4, n−2. UsingAlgorithm 3there exist two edgese₁, e₂such thate₁ ∈P ore₂∈Pis on the boundary ofFm, nfacingR2m− 4, n−2, seeFigure 5a. Hence by combining a Hamiltonian pathPand Hamiltonian cycles in R2,2n−2andR2m−2,2n−2andv1, v2, a Hamiltonian path betweensandtis obtained, see Figures5band5c.

Subcase2.2.2 m >3. Form 4, let four verticesv₁, v₂, v₃, v₄ be inR2m−4, n−2. Using Algorithm 3there exist four edgese1, e2, e3, e4 such thate1, e2 ∈ P,e1, e4 ∈ P,e3, e4 ∈ P or e₂, e₃ ∈ Pare on the boundary ofFm, nfacingR2m−4, n−2seeFigure 6a. Therefore, by mergingv₁, v₂andv₃, v₄to these edges and Hamiltonian cycles inR−Fwe obtain a Hamiltonian path forR, seeFigure 6b. For other values ofm, the proof is similar to that of m4.

Similar toF-alphabet grid graphs, Hamiltonian paths can be found inE-alphabet grid graphs forn3.

CombiningLemma 3.2andTheorem 2.1the following corollary is trivial.

Corollary 3.3. LetAm, nbe anL-alphabet,C-alphabet,F-alphabet, orE-alphabet grid graph andR be the smallest rectangular grid graph that includesA. IfAm, n, s, tis Hamiltonian, thensandt must be color compatible inR.

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e1 e2

v1 v2

e3 e4

v3 v4

a

s

t

b Figure 6:A Hamiltonian path inR7,11.

Therefore, the color compatibility of s and t in R is a necessary condition for Lm, n, s, t,Cm, n, s, t,Fm, n, s, t, andEm, n, s, tto be Hamiltonian.

Deﬁnition 3.4. The length of a path in a grid graph means the number of vertices of the path. In any grid graph, the length of any path between two same-colored vertices is odd and the length of any path between two diﬀerent-colored vertices is even.

Lemma 3.5. LetR2m−2, nandRm,5n−4be a separation ofLm, nsuch that three verticesv, w, anduare inR2m−2, nwhich are connected toRm,5n−4. Assume thatsandtare two given vertices ofLands^′wandt^′t, ifs∈R2m−2, nlets^′s. Iftx> m 1andR2m−2, n, s^′, t^′ satisﬁes condition (F3), thenLm, ndoes not have any Hamiltonian path betweensandt.

Proof. Without loss of generality, let s and t be color compatible. Since m, n ≥ 3 and by Theorem 2.1a rectangular grid graph does not have a Hamiltonian path only in condition F3, so it suﬃces to prove the lemma for the casen 3. Assume thatR2m−2, nsatisﬁes condition F3. We show that there is no Hamiltonian path in Lm, n between s and t.

Assume to the contrary thatLm, nhas a Hamiltonian pathP. Sincen3 there are exactly three verticesv, wanduinR2m−2, nwhich are connected toRm,5n−4, as shown in Figure 7a. Then the following cases are possible.

Case 1. t∈R2m−2, nands /∈R2m−2, n. The following subcases are possible for the Hamil- tonian pathP.

Subcase 1.1. The Hamiltonian pathPofLm, nthat starts fromsmay enter toR2m−2, nfor the ﬁrst time through one of the verticesv, w, oru, pass through all the vertices ofR2m−2, n and end att, seeFigure 7b. This case is not possible because we assumed thatR2m−2, n satisﬁesF3, in this caset^′tands^′w.

Subcase 1.2. The Hamiltonian pathPofLm, nmay enter toR2m−2, n, pass through some vertices of it, then leave it and enter it again and pass through all the remaining vertices of it and ﬁnally end att. In this case, two subpaths ofPwhich are inR2m−2, nare calledP1and P2,P1fromvtouvtoworutowandP2fromwtotutotorvtot. This case is not also

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R(m,5n−4)

v w u

R(2m−2, n)

a

v w u

s^′ t^′ s

t

b Figure 7:A separation ofL4,3.

possible because the size ofP1is oddevenand the size ofP2is evenodd, then|P1 P2|is odd whileR2m−2, nis even, which is a contradiction.

Case 2s, t∈R2m−2, n. The following cases may be considered.

Subcase 2.1. The Hamiltonian pathP ofLm, nwhich starts fromsmay pass through some vertices ofR2m−2, n, leaveR2m−2, natvoru, then passes through all the vertices of Rm,5n−4and reenter toR2m−2, natwgo touorvand pass through all the remaining vertices ofR2m−2, nand end att. In this case by connectingvorutowwe obtain a Hamil- tonian path fromstotinR2m−2, n, which contradicts the assumption thatR2m−2, n satisﬁesF3.

Subcase 2.2. The Hamiltonian pathPofLm, nwhich starts fromsmay leaveR2m−2, nat voru, then pass through all the vertices ofRm,5n−4and reenter toR2m−2, natuor vgo tow and pass through all the remaining vertices ofR2m−2, nand end att. In this case, two parts ofP reside inR2m−2, n. The partP1starts fromsends atvoru, and the partP₂starts fromuorvends att. The size ofP₁is even and the size ofP₂is odd while the size ofR2m−2, nis even, which is a contradiction.

Subcase 2.3. Another case that may arise is that the Hamiltonian pathPofLm, nstarts from sleavesR2m−2, natwand reentersR2m−2, natvoruand then goes tot. But in this case vertexuorvcannot be inP, which is a contradiction. Thus, the proof ofLemma 3.5is completed.

Lemma 3.6. LetLm, nandR2m−2, nbe a separation ofCm, nsuch that three verticesv1,w1, andu1are inR2m−2, nwhich are connected toLm, nandx∈Lm, nis a adjacent vertex tow1. Assume thatsandtare two given vertices ofC. Ift∈Lm, nands∈R2m−2, n, letsxand s^′w1,t^′s, respectively. IfLm, n, s, tdoes not have a Hamiltonian path orR2m−2, n, s^′, t^′ satisﬁes condition (F3), thenCm, ndoes not have a Hamiltonian path betweensandt.

Proof. The proof is similar to the proof ofLemma 3.5, for more details seeFigure 8.

Lemma 3.7. LetLm, nandR2m−4, nbe a separation ofFm, nsuch that three verticesv1,w1, andu1are inR2m−4, nwhich are connected toLm, nandx∈Lm, nis an adjacent vertex to

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L(m, n)

v1

w1

u1

s t

R(2m−2, n)

a

L(m, n)

x s

t

w1

R(2m−2, n) s^′

t^′ s

b

x L(m, n)

s

t w1

s^′ t^′ R(2m−2, n) s

c

Figure 8:C-alphabet grid graphs in which there is no Hamiltonian path betweensandt.

L(m, n)

s^′ t^′ v1

w1

u1

x

s

t t

a

L(m, n)

w1

x s

s

t

s^′

t^′

b

s

t v w u

L(m, n)

s^′ t^′

c

Figure 9:F-alphabet grid graphs in which there is no Hamiltonian path betweensandt.

w1. Assume thatsandtare two given vertices ofF. Ifs, t∈R2m−4, nlets^′sandt^′t; ifs(ort)

∈R2m−4, nlets^′w1andt^′s(ors^′w1andt^′t); ift(ors)∈Lm, nletsxts, sx ort x. IfLm, n, s, tdoes not have a Hamiltonian path orR2m−4, n, s^′, t^′satisﬁes the condition (F3), thenFm, ndoes not have a Hamiltonian path betweensandt.

Proof. By a similar way as used in theLemma 3.5we can prove this lemma; for more details seeFigure 9.

Lemma 3.8. LetFm, nandR2m−2, norCm, nandR2m−4, nbe a separation ofEm, n and s and t be two given vertices inEm, n. IfFm, n, s, t or Cm, n, s, t does not have a Hamiltonian path, thenEm, ndoes not have a Hamiltonian path betweensandt.

Proof. The proof is similar to the proof ofLemma 3.5, for more details seeFigure 10.

From Corollary 3.3 and Lemmas 3.5, 3.6,3.7, and 3.8, a Hamiltonian path problem PLm, n, s, t is called acceptable if s and t are color-compatible and R2m −2, n, s^′, t^′

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F(m, n)

R(2m−2n, n) s

t

a

C(m, n)

s t

b

Figure 10:E-alphabet grid graphs in which there is no Hamiltonian path betweensandt.

does not satisfy the condition F3; PCm, n, s, t is called acceptable if PLm, n, s, t is acceptable andR2m−2, n, s^′, t^′does not satisfy the conditionF3;PFm, n, s, tis called acceptableifPLm, n, s, tis acceptable andR2m−4, n, s^′, t^′does not satisfy the condition F3;PEm, n, s, tis calledacceptableifPFm, n, s, tandCm, n, s, tare acceptable.

Theorem 3.9. Let Am, nbe an L-alphabet,C-alphabet,F-alphabet or E-alphabet grid graph. If Am, n, s, tis Hamiltonian, thenPAm, n, s, tis acceptable.

Now, we show that all acceptable Hamiltonian path problems have solutions by introducing algorithms to find Hamiltonian pathssufficient conditions. Our algorithms are based on a divide-and-conquer approach. In the dividing phase we use two operations stirp and split which are defined in the following.

Deﬁnition 3.10. A subgraphS of anL-alphabet, C-alphabet,F-alphabet orE-alphabet grid graph A strips a Hamiltonian path problem PAm, n, s, t, if all of the following four conditions hold:

1Sis even sized and:

iS is a rectangular grid graph; where Am, n is a L-alphabet grid graph Lm, n, a C-alphabet grid graph Cm, n and an E-alphabet grid graph Em, n;

iiS is a L-alphabet graph Lm, n, a rectangular graph R2m−4, n or three rectangular grid graphsR1−R3; whereAm, nis aF-alphabet grid graph.

2SandA−Sis a separation ofA;

3s, t∈A−S;

4PA−S, s, tis acceptable.

Deﬁnition 3.11. Let PAm, n,s,t be a Hamiltonian path problem and p, q be an edge of A, whereAm, n is anL-alphabet,C-alphabet, or F-alphabet grid graph. Then we say p, qsplitsAm,n,s,tif there exists a separation of

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S

L−S

s t

a

S

L−S

s t

b

C−S

S

s t

c Figure 11:a,bA strip ofL3,3,ca strip ofC3,3.

Rq

p q Rp

s

t

a

p q Rp

Rq

s

t

b

p

q R_p

Lq

s

t

c Figure 12:a,bA split ofL3,3,ca split ofC3,3.

1LintoR_pandR_qsuch that

is, p∈R_pandPR_p, s, pis acceptable, iiq, t∈R_qandPR_q, q, tis acceptable.

2Fresp.,CintoRpandLqorRqandLpresp.,LqandRpsuch that

is, p∈RpandPRp, s, pis acceptableors, p∈LpandPLp, s, pis acceptable, iiq, t∈LqandPLq, q, tis acceptableorq, t∈RqandPRq, q, tis acceptable.

In the following, we describe for each alphabet class how the solutions of the subgraphs are merged to construct a Hamiltonian path for the given input graph.

3.1. Hamiltonian Paths inL-Alphabet Grid GraphsLm, n

Since anL-alphabet graphLm, nmay be partitioned into two rectangular grid graphs, then the possible cases for verticessandtare as follows.

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L(m, n)

s

t

R(2m−4, n)

a

L(m, n)

s

t

R(2m−4, n)

b

s t

R(2m−2, n) F(m, n)

c Figure 13:a,bA strip ofF4,3andF3,3,ca strip ofE3,3.

L(m, n)

s t

R(2m−4, n)

a

s t

R1

R2

R3 R4

b

s t

R1

R2

R3 R4

c Figure 14:A strip ofF3,3, whens, t∈R2m−4, n.

Case 1s, t∈ L−S. Assume thatL−Shas a Hamiltonian pathP by the algorithm of12, where L−S is a rectangular gird graph. Since S is an even-sized rectangular gird graph, then it has Hamiltonian cycle byLemma 2.2; seeFigure 11a. Therefore, a Hamiltonian path forLm, n, s, tcan be obtained by mergingPand the Hamiltonian cycle ofSas shown in Figure 11b.

Case 2s∈Rpandt∈Rq. In this case, we construct Hamiltonian paths inRpandRqbetween s, pandq, t, respectively; seeFigure 12a. Then a Hamiltonian path forLm, n, s, tcan be obtained by connecting two verticespandqas shown inFigure 12b.

3.2. Hamiltonian Paths inC-Alphabet Grid GraphsCm, n

Similar toL-alphabet graphs since aC-alphabet graphCm, nmay be partitioned into aL- alphabet graphLm, nand a rectangular grid graphR2m−2, n, then the possible cases for verticessandtare as follows.

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Case 1s, t∈C−S. Assume thatC−Shas a Hamiltonian pathPby theAlgorithm 1, where C−Sis anL-alphabet gird graphLm, n. Hence, a Hamiltonian path forCm, n, s, tcan be obtained by mergingPand the Hamiltonian cycle ofSas shown inFigure 11c.

Case 2s∈Rpandq∈Lq. A Hamiltonian path can be found asFigure 12c.

3.3. Hamiltonian Paths inF-Alphabet Grid GraphsFm, n ForF-alphabet grid graphs, we consider the following cases.

Case 1s, t ∈ F−S, whereF−SisLm, n. In this case, we construct a Hamiltonian path betweensandtinLm, nby theAlgorithm 1. SinceSis an even-sized rectangular grid graph R2m−4, n, then it has a Hamiltonian cycle byLemma 2.2.

Case 2s, t∈F−S, whereF−SisR2m−4, n. We construct a Hamiltonian path betweens andtinR2m−4, nby the algorithm in12. SinceSis an even-sizedL-alphabet grid graph Lm, n, then it has a Hamiltonian cycle byLemma 2.3.

In two cases, by combining the Hamiltonian cycle ofSand the Hamiltonian path ofL orR2m−4, n, a Hamiltonian path betweensandtforFm, nis obtained, see Figures13a and13b.

Case 3Fm, nis odd sized ands, t∈F−S, whereF−SisR2m−4, n. In this case,sandt must be white. SinceR2m−4, nis even sized, thensandtare not color compatible inR2m−

4, n, seeFigure 14a. Therefore, we repartitionFm, ninto four rectangular grid graphsR1, R2, R3, and R4, such that s, t ∈ VR1 andR2,R3, and R4 are even sized, seeFigure 14b where the dotted lines represent the strip. In this case, by combining the Hamiltonian cycles inR2,R3, andR4and the Hamiltonian path ofR1, a Hamiltonian path between sandtfor Fm, n, s, tis obtained, seeFigure 14c.

Case 4 s ∈ R_p and t ∈ L_q or s ∈ L_p and q ∈ R_q. A Hamiltonian path can be found as Figure 15.

3.4. Hamiltonian Paths inE-Alphabet Grid GraphsEm, n

A Hamiltonian path for anE-alphabet grid graphEm, ncan be found using striping. So we stripEm, nsuch thats, t∈E−S, whereE−SisF-alphabetFm, norC-alphabetCm, n. In this case, we construct a Hamiltonian path in Fm, n or Cm, n by Algorithm 3 or 4. Since S is an even-sized rectangular grid graph R2m−2, n or R2m−4, n, then it has a Hamiltonian cycle by Lemma 2.2. By combining the Hamiltonian cycle of Sand the Hamiltonian path ofForC, a Hamiltonian path betweensandtforEm, nis obtained, see Figure 13c.

Thus, we have the following lemmas.

Lemma 3.12. LetAm, n, s, tbe an acceptable Hamiltonian path problem, andSstrips it, where Am, nis anL-alphabet,C-alphabet,F-alphabet, orE-alphabet grid graph. IfA−Shas a Hamiltonian path betweensandt, thenAm, n, s, thas a Hamiltonian path betweensandt.

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s t L(m, n)

R(2m−4, n)

a

L(m, n)

s t

p q

b

s t

p q

c Figure 15:A split ofF3,3.

procedureLHamiltonianPathLm, n, s, t 1: ifLcan be strippedthen

2: letSbe a strip ofL, whereSis an even-sized rectangular grid graph 3: P←RHamiltonianPathL−S, s, t

4: D←HamiltonianCycleS 5: returnMergeStripP, D, s, t 6: else

7: letLbe split toRpandRq

8: P1←RHamiltonianPathRp, s, p 9: P2←RHamiltonianPathRq, q, t 10: returnMergeSplitP1, P2, p, q 11: end if

Algorithm 1:The Hamiltonian path algorithm forL-alphabet grid graphs.

Lemma 3.13. Letp, qbe an edge which splitsPAm, n, s, t. If

i R_p, s, pandR_q, q, t, whereAm, nisL-alphabet grid graphLm, n, ii Lq, q, tandRp, s, p, whereAm, nisC-alphabet grid graphCm, n,

iii R_p, s, pandL_q, q, torR_q, q, tandL_p, s, p, whereAm, nisF-alphabet grid graph Fm, n,

have a Hamiltonian path betweens, pandq, t, thenAm, nalso has a Hamiltonian path between sandt.

Since all the proofs presented in this section were constructive, they give us algorithms for finding a Hamiltonian path inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs. The pseudo-codes of the algorithms are given in Algorithms1,2,3, and4. In these algorithms, HamiltonianCycle is the procedure that finds the Hamiltonian cycle of a strip based on Lemma 2.2 or Lemma 2.3, MergeStrip is a procedure that merges a path and a cycles,MergeCyclesis a procedure that combine two Hamiltonian cycles using two parallel edges ofe1 and e2,MergeSplitis a procedure that connects two paths by simply adding an edge between their end vertices p and q, and R HamiltonianPathis a procedure that finds

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procedureCHamiltonianPathCm, n, s, t 1: ifCcan be strippedthen

2: letSbe a strip ofC, whereSis an even-sized rectangular grid graphR2m−2, n 3: P←LHamiltonianPathC−S, s, t

7: letCbe splitRpandLq

8: P1←RHamiltonianPathRp, s, p 9: P2←LHamiltonianPathLq, q, t 10: returnMergeSplitP1, P2, p, q 11: end if

Algorithm 2:The Hamiltonian path algorithm forC-alphabet grid graphs.

procedureFHamiltonianPathFm, n, s, t

1: ifFcan be stripped, whereFm, nis not odd sized ands, t /∈R2m−4, nthen 2: letSbe a strip ofF

3: P←F−SHamiltonianPathF−S, s, t

4: /^∗F−Sis aL-alphabet grid graphLm, nor a rectangular grid graphR2m−4, n

∗/

8: ifFcan be stripped, whereFm, nis odd sized ands, t∈R2m−4, nthen 9: letSbe a strip ofF, whereSis three rectangular grid graphsR2, R3andR4

10: P←RHamiltonianPathR1, s, t 11: c1←HamiltonianCycleR3 12: c2←HamiltonianCycleR4 13: C←MergeCyclesc1, c2 14: D←HamiltonianCycleR2 15: returnMergeStripP, D, C, s, t 16: else

17: letFbe split toRpandLqorLpandRq 18: P1←RHamiltonianPathRp, s, p 19: P2←LHamiltonianPathLq, q, t 20: returnMergeSplitP1, P2, p, q 21: end if

22: end if

Algorithm 3:The Hamiltonian path algorithm forF-alphabet grid graphs.

a Hamiltonian path in a rectangular grid graph by the algorithm in 12. The algorithm ﬁrst checks if the input graphAcan be stripped, then stripsAbyS, and recursively ﬁnds a Hamiltonian path ofA−S. Otherwise, ifAcan be split, then splitsAintoRp andRq Lq

andRporRqandLp, and recursively ﬁnds Hamiltonian paths ofRpandRqLqandRporRq

andL_p. Then these two Hamiltonian paths are merged into a single path, whereAm, nis anL-alphabet,C-alphabet,F-alphabet, orE-alphabet grid graph.

FromTheorem 3.9and Lemmas3.12and3.13the following theorem holds.

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procedureEHamiltonianPathEm, n, s, t

1: letSbe a strip ofE, whereSis an even-sized rectangular grid graphR2m−2, nor R2m−4, n

2: P←E−SHamiltonianPathE−S, s, t

3: /^∗E−Sis aF-alphabet grid graphFm, nor aC-alphabet grid graphCm, n^∗/ 4: D←HamiltonianCycleS

5: returnMergeStripP, D, s, t

Algorithm 4:The Hamiltonian path algorithm forE-alphabet grid graphs.

Theorem 3.14. LetAm, nbe anL-alphabet,C-alphabet,F-alphabet, orE-alphabet grid graph, and sandtbe two distinct vertices of it.Am, nhas Hamiltonian path if and only ifPAm, n, s, tis acceptable.

Theorem 3.14provides necessary and suﬃcient conditions for the existence of Hamil- tonian paths inL-alphabet,C-alphabet,F-alphabet andE-alphabet grid graphs.

CombiningTheorem 3.14and the Algorithms1,2,3, and4we arrive at the main result.

Theorem 3.15. InL-alphabet,C-alphabet,F-alphabet orE-alphabet grid graphs, a Hamiltonian path between any two verticessandtcan be found in linear time.

Proof. The algorithms divide the problem into some rectangular grid graphs inO1. Then we solve the subproblems in linear time using the linear time algorithm in12. Then the results are merged in timeO1using the method proposed in12.

4. Conclusion and Future Work

In this paper, we presented linear time algorithms for ﬁnding a Hamiltonian path inL-alphabet, C-alphabet, F-alphabet, and E-alphabet grid graphs between any two given vertices.

Since the Hamiltonian path problem is NP-complete in general grid graphs, it remains open if the problem is polynomially solvable in solid grid graphs.

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