There is but one PDS in \mathbb{Z}^{3} inducing just square components
aa r X i v : . [ m a t h . C O ] F e b There is but one PDS in Z inducing just square components Italo J. Dejter, Luis R. Fuentes
University of Puerto RicoRio Piedras, PR 00936-8377 e-mail: [email protected], [email protected] andCarlos A. Araujo
Universidad del Atl´anticoBarranquilla, Colombia e-mail: [email protected]
Abstract
It is known that in the unit distance graph of the lattice Z ⊂ R thereexists a dominating set S with 4-cycles as sole induced components and eachvertex of Z \ S having a unique neighbor in S . We show S is unique. Keywords: perfect dominating sets; unit distance graph; integer lattice.
Primary 05C69; Secondary94B25.
1. PERFECT DOMINATING SETS, (PDS s)Let Γ = (
V, E ) be a graph and let S ⊂ V . The closed neighborhood of a vertex θ ∈ V in Γ is denoted N [ θ ]. Let [ S ] be the subgraph of Γ induced by S . Theinduced components of S , namely the connected components of [ S ] in Γ, aresaid to be the components of S . Several definitions of perfect dominating sets ingraphs are considered in the literature [19, 21]. We work with the following one[24] denoted with the short acronym PDS, to make a distinctive difference: S is a PDS of Γ ⇔ each vertex of V \ S has a unique neighbor in S .This definition (of PDS) differs from that of a ‘perfect dominating set’ as in[16, 18, 23] (that for us is a stable PDS coinciding with the perfect code of [4] orwith the efficient dominating set of [3, 11, 19]). With our not necessarily stableefinition of perfect dominating set, denoted PDS, our main result, stated belowas Theorem 1.1, has a narrowing spirit as that of Theorem 2.6 of just cited [23].Let 0 < n ∈ Z . The following graphs are considered. The unit distance graphΛ n of the n []-dimensional integer lattice Z n ⊂ R n has vertex set Z n and exactlyone edge between each two vertices if and only if their Euclidean distance is 1.An n - cube is the cartesian graph product Q n = K (cid:3) K (cid:3) · · · (cid:3) K of precisely n copies of the complete graph K . In particular, a 2-cube Q is a square, that isa 4-cycle. A grid graph is the cartesian graph product of two path graphs.Our definition of a PDS S allows components of S in Γ which are not isolatedvertices. For example: (a) tilings with generalized Lee r -spheres, for fixed r with1 < r ≤ n in Z (e.g., crosses with arms of length one if r = n ), furnish Λ n withPDS s whose components are r -cubes [15], including that of our Theorem 1.1,below; (It is most remarkable that r = n ⇔ n ∈ { r − , r −
1; 0 < r ∈ Z } [6]); (b) total perfect codes [1, 22], that is PDS s whose components are copies of K = P in the Λ n s and grid graphs; (these appear as diameter perfect Lee codes [14, 20]); (c) PDS s in n -cubes [5, 7, 8, 10, 12, 24], where 0 < n ∈ Z , includingthe perfect codes of [13]; (d) PDS s in grid graphs [8, 22].
Theorem 1.1.
There is only one PDS in Λ whose components are 4-cycles. This is proved as Theorem 4.1 once some auxiliary notions are presented.2. INDUCED COMPONENTSThe distance d ( u, v ) between two vertices u and v of Λ n is defined as the minimumlength of any path connecting u and v . The following is an elementary extensionof a result of [24] for n -cubes. Theorem 2.1.
Let S be a PDS in Λ n . Let J S be a set of indices j for thecorresponding components S j of S . Each S j is a cartesian graph product ofconnected subgraphs of Λ . Thus, if such S j is a finite subgraph Θ of Λ n , then S j is of the form P i j (cid:3) P i j (cid:3) · · · (cid:3) P i jn , where P i jk is a path of length i jk − ≥ , for k = 1 , . . . , n . A PDS in Λ n whose components are all isomorphic to a fixed finite graph Θ(as in Theorem 2.1) is called a PDS[Θ]. If no confusion arises, n -tuples repre-senting elements of Z n are written with neither commas nor external parentheses.We denote 00 . . . O , 10 . . . e , 010 . . . e , . . . , 00 . . . e n .At the end of Section 6 of [15] (in the original setting of item (a) above inSection 1), all the indices i jk of our Theorem 2.1 are shown to be less than 2.2. LATTICE-LIKE DOMINATING SETSLet Θ = ( V, E ) be a finite subgraph of Λ n and let z ∈ Z n . Then Θ + z denotesthe graph Θ ′ = ( V ′ , E ′ ), where V ′ = V + z = { w ; there is v ∈ V, w = v + z } and uv ∈ E ⇔ ( u + z )( v + z ) ∈ E ′ . Let S be a PDS[Θ] and let a copy D of Θ be acomponent of S . Then S is said to be lattice-like if there is a lattice L (that is,a subgroup L of Z n ) so that D ′ is a component of S if and only if there is z ∈ L with D ′ = D + z . Examples above ([15, 6, 14, 20]) are lattice-like.If S is a PDS[Θ] with Θ = ( V, E ), then S can be seen as a tiling of Z n bythe induced subgraph Θ ∗ of Λ n on the set V ∗ = { v ∈ Z n ; d ( v, V ) ≤ } . Thus, alattice-like tiling will be understood in the same way as a lattice-like PDS. Weneed the following form of Theorem 6 [20] for the proof of Theorem 4.1. Recallthat given a graph G , the distance d ( v, H ) between a vertex v of G and a subgraph H of G is the shortest distance between v and the vertices of H . Theorem 3.1.
Let Θ be a subgraph of Λ n . Let Θ ∗ be an induced supergraph of Θ in Λ n such that a vertex v is in Θ ∗ if and only if d ( v, Θ) ≤ . Let D = ( V, E ) be a copy of Θ ∗ that contains vertices O, e , . . . , e n . Then, there is a lattice-likePDS [Θ] if and only if there exists an abelian group G of order | V | and a groupepimorphism Φ : Z n → G such that the restriction of Φ to V is a bijection.
4. THE PROOF
Theorem 4.1.
There do not exist non-lattice-like PDS [ Q ] s in Λ . In addition,there exists exactly one lattice-like PDS [ Q ] in Λ . Figure 1. Θ ⊂ Θ ∗ ⊂ Θ ′ , and the case of one corner of Θ ′ − Θ ∗ in S igure 2. Instances of no corners of Θ ′ − Θ ∗ in S , (a ) and (a ) roof. Theorem 8 [15] insures the existence of a PDS[ Q ] in Λ . In fact, the con-nected components of such PDS[ Q ] are the generalized Lee spheres S , , insidethe corresponding generalized Lee spheres S , , (in their inductive constructionin Section 1 [15]) that form the lattice tiling Λ , (in the notation of [15]) insuredby that Theorem 8. According to the theorem, this Λ , has generator matrix (asdefined in Section 3 [15]): (1) In terms of Theorem 3.1, the generator matrix (1) corresponds to the groupepimorphism Φ : Z → G = Z given by Φ( e ) = 2; Φ( e ) = 5 and Φ( e ) = 6where Φ is obtained first by multiplying the matrix (1) by an unknown vector andthen solving the corresponding system of equations mod 20. To see that this isthe only PDS[ Q ] in Λ , we note that there are only two possible abelian groups G for the epimorphism Φ, namely: G = Z and G = Z × Z × Z . It can beeasily checked [17] that (a) there are just 32 epimorphism from Z onto G = Z and none from Z onto G = Z × Z × Z ; (b) every possible assignment forΦ( e ), Φ( e ) and Φ( e ) has order 10, 10 and 4, respectively, in Z . As a result,all 4-cycles induced by each lattice-like PDS[ Q ] associated (via Theorem 3.1) toa corresponding of these 32 epimorphisms are placed in the same way in Λ . Eachsuch lattice-like PDS[ Q ] in Λ is equivalent to the one obtained via matrix (1).Assume there is a non-lattice-like PDS[ Q ] S in Λ so that the componentsof [ S ] are 4-cycles Q ; let Θ = Θ be such a component. We may assume thatΘ has vertices O , e , e , e + e . The graph Θ ∗ = Θ ∗ is contained in a graph Θ ′ isomorphic to P (cid:3) P (cid:3) P as on the left of Figure 1, where Θ has its edges thickblack, the rest of Θ ∗ has them red and the rest of Θ ′ has them green, thick forthe paths between the eight corners (vertices of degree 3 in Θ ′ : − e − e ± e ,2 e − e ± e , − e + 2 e ± e , 2 e + 2 e ± e ) and thin for the rest. The realizationof Θ ∗ in R has convex hull containing tightly Θ ′ . Similar colors and tracesare used in the representations in Figures 2-14, where: (I) The red thin-tracelines un Figures 2-3 and 4-5 represent edges incident to vertices in subgraphs Θ ∗ (red thick-trace edges) involved in our arguments by contradiction, indicated byquestion marks (?); (II) yellow squares in Figures 7-14 indicate where to pasteaccordingly (the front of) the top and (the back of) the lower parts of each figureto obtain a continuation of the lattice representation in each figure; (III) edgesnot mentioned in (I) or (II) are traced in dashed green color.Assume no vertex of Θ ′ − Θ ∗ is in S . By symmetry there is a 1-factor F inΘ ′ − Θ ∗ each of whose edges has an endvertex / ∈ V (Θ ′ − Θ ∗ ) dominated by a vertexin a 4-cycle induced by S . In each case we will reach a contradiction: F is either5s in case (a) or (b) below, depending on the feasible dispositions of four edgesof F over the four maximal paths of length 2 between the eight corners of Θ ′ ,namely either with their eight endvertices having convex hull tightly containinga copy of P (cid:3) P (cid:3) P (say convex hull [ − , × [ − , × [ − , {− } × [ − , × [ − ,
0] and { } × [ − , × [0 ,
1] appear,not leading to a total convex hull as above), that we have respectively either asthe four edges f , f , f , f , for (a), or as the four edges f , f , f , f for (b).These instances are: (with (a) further subdivided into subcases (a ) and (a ),below) (a) (Figure 2, top) The edges of F are: f =( − e − e , − e − e − e ) ,f =( − e − e , e − e − e ) , f =( − e + e , − e + e + e ) ,f =( 2 e − e , e − e − e ) , f =( − e − e , − e + e − e ) ,f =(2 e + e , e + e + e ) ,f =( 2 e − e , e + e − e ) ,f =( e − e + e , e − e + e ) , f =(2 e + e , − e +2 e + e ) ,f =( − e + e , − e − e + e ) , f =( e +2 e + e , e +2 e + e ) ,f =( − e +2 e , − e +2 e − e ) ,f =( 2 e +2 e , e +2 e − e ) , f =(2 e − e , e +2 e − e ) . We may take step by step either option (a ) or option (a ) below (where, insteadof saying that a vertex v is dominated by an endvertex of an edge f , we simplysay that v is dominated by f , or that v ∈ ( f )), with ( f ) representing the set ofvertices dominated by the endvertices of f ): (a ) The first eight edges in (a) have each an endvertex dominated by avertex in a 4-cycle. The involved 4-cycles contain the following edges: • f − e (the translation of f via the vector − e ), • f + e (forced, since f − e dominates − e ∈ ( f − e )), • f − e (forced, since f − e contains − e − e ∈ ( f − e )), • f − e (forced, since f − e dominates − e − e − e ∈ ( f − e )), • f + e (forced, since f − e contains 2 e − e − e ∈ ( f − e )), • f + e (forced, since f + e dominates 3 e ∈ ( f + e )), • f − e (forced, since f + e contains 3 e − e ∈ ( f + e )) and • f + e (forced, since f + e dominates − e + 2 e + 2 e ∈ ( f + e )).Now, there is no way for the edge f to be dominated by a copy of K externalto Θ ′ − Θ ∗ (since f + e contains e + 3 e + e ∈ ( f + e ) while f + e contains2 e + 2 e + 2 e ∈ ( f + e )), a contradiction. (a ) (Figure 2, bottom) The edges f , f and f have each one endvertexdominated by a vertex in a 4-cycle containing the respective edges f − e , f − e (edge pair not contemplated in case (a )) and f − e (forced, since f − e contains vertex − e ∈ ( f − e )). But then only one of f and f must bedominated by f − e or f − e , while the remaining one must be dominatedby f − e or f − e , which produces a contradiction since f − e ∈ ( f − e ),and f − e ∈ ( f − e ). 6 igure 3. Instance of no corners of Θ ′ − Θ ∗ in S , case (b) igure 4. The two cases of three corners of Θ ′ − Θ ∗ in S (b) (Figure 3) The edges of F are: f =( − e − e , − e − e − e ) ,f =( − e +2 e , − e +2 e − e ) , f =( − e + e , − e − e + e ) ,f =( 2 e − e , e +2 e − e ) , f =( − e + e + e , − e +2 e + e ) ,f =( 2 e + e , e +2 e + e ) ,f =( 2 e +2 e − e , e + e − e ) ,f =( − e − e , e − e − e ) , f =( 2 e +2 e , e +2 e + e ) ,f =( − e + e , e − e + e ) , f =( 2 e − e , e − e − e ) ,f =( 2 e − e , e − e + e ) ,f =( − e − e , − e + e − e ) , f =( 2 e + e , e + e + e ) . We may assume step by step that the first ten edges of F have each anendvertex dominated by the copy of K containing respectively: • f − e , • f + e (forced, since f − e contains e − e ∈ ( f − e )), • f − e (forced, since f + e contains e + 2 e − e ∈ ( f + e )), • f + e (forced, since f − e contains 2 e − e ∈ ( f + e )), • f − e (forced, since f + e contains 3 e − e ∈ ( f + e )), • f + e (forced, since f + e contains 3 e + e ∈ ( f + e )), • f + e (forced, since f − e contains 2 e + 2 e − ∈ ( f − e )), • f + e (forced, since f + e contains 3 e + 2 e ∈ ( f + e )), • f − e (forced, since f + e contains 3 e − e ∈ ( f + e )) and • f − e (forced, since f − e contains e − e − e ∈ ( f − e )).Now, f does not have an endvertex dominated by any copy of K in the presenceof the previous forced dominations of copies of K (since f − e dominates {− e , − e } ⊂ ( f − e ) while f + e contains 2 e − e ∈ ( f + e )).If just one or three corners of Θ ′ (in this second case, for corner distance tripleeither (3 , ,
6) or (3 , , S , the remaining vertices of Θ ′ − Θ ∗ forms no8-factor F , contradicting the existence of S . (Figure 1, right, and Figure 4). Inthe case of one corner, let this corner be θ = − e − e − e , which dominates θ + e , θ + e and θ + e . Then F must contain: f =( e − e − e , e − e − e ) ,f =( − e + e , − e + e + e ) , f =( − e + e − e , − e +2 e − e ) ,f =(2 e − e , e − e + e ) , f =( − e +2 e , − e +2 e + e ) ,f =( − e − e + e , − e + e ) . Now, F should also contain f = ( e − e + e , e − e + e ), with its terminal vertexalready present in f , a contradiction. With three corners and distance triple(3 , , θ , θ = 2 e − e − e and θ = 2 e + 2 e − e . Then F must contain f = ( − e + e − e , − e +2 e − e ) and f = ( − e +2 e − e , e − e )that have a vertex in common, a contradiction. With distance triple (3 , , θ , θ and θ ′ = θ + 2 e . Then F must contain f as aboveand f ′ = f + 2 e , leaving vertex − e + 2 e not in F , another contradiction. Figure 5. The case of two corners of Θ ′ − Θ ∗ in S at distance 3 We will rule out the cases of only two corners of Θ ′ being in S . If the two areat distance 3 (Figure 5) they may be taken up to symmetry as θ = − e − e − e and θ = 2 e − e − e . In Θ ′ − Θ ∗ − N [ θ ] − N [ θ ], we note a unique 1-factor F ,formed by edges f = ( − e − e + e , − e + e ), f = ( e − e + e , e − e + e ), f =(2 e + e , e + e + e ), f = (2 e +2 e , e +2 e + e ), f = (2 e + e , e +2 e + e ), f = (2 e − e , e + 2 e − e ), f = ( − e + 2 e − e , − e + e − e ), etc. The copies9 igure 6. The two cases of two corners of Θ ′ − Θ ∗ in S at distance 5 igure 7. Four corners, instance (A), case (a) K containing f , . . . , f can be taken dominated, by symmetry and forcedly,by the copies of K containing f − e , f + e , f + e , f + e , f + e and f − e respectively. The 4-cycle induced in S that contains θ , also contains forcedlythe vertices θ − e , θ − e − e and θ − e . But then, f cannot be dominatedin S , a contradiction.Now, assume that the two corners are at distance 5, (Figure 6). They maybe taken up to symmetry as θ = − e − e − e and θ = 2 e − e + e . InΘ ′ − Θ ∗ − N [ θ ] − N [ θ ] we observe a unique 1-factor F , formed by edges f =( − e + e − e , − e + 2 e − e ), f = ( e − e − e , e − e − e ), f = (2 e + e , e + 2 e + e ), f = (2 e + 2 e , e + 2 e − e ), f = (2 e − e , e + e − e ), f = ( − e + e , − e + e + e ), f = (2 e − e , e + 2 e − e ), etc. If the edge( θ , θ − e ) is in S , then f − e , f − e , f + e , f + e and f − e dominaterespectively f , f f , f and f . But then f cannot be dominated in S , acontradiction. So, F forces the 4-cycle with vertices θ , θ − e , θ − e − e and θ − e to be in S . In this case, the copies of K associated to f , f , f and f are dominated respectively by the copies of K containing f − e , f − e , f + e and f + e . It follows that f cannot be dominated by an edge at distance 1from it in Λ − Θ ′ , a contradiction.It is easy to see that two corners at distance 6 or 8 do not allow even thedefinition of a 1-factor F in Θ ′ − Θ ∗ minus the two corners and their neighbors.We pass to consider the different cases of four corners of S in Θ ′ − Θ ∗ . Thecase of S having three corners on the affine plane < e , e > − e and one cornerin the affine plane < e , e > + e , or viceversa, is readily seen to lead to no1-factor F in Θ ′ − Θ ∗ minus these corners and their neighbors. Else, either: Instance (A) : If the four corners in S are θ = − e − e − e , θ = 2 e − e − e , θ = − e +2 e + e and θ = 2 e +2 e + e , then a 1-factor F of Θ ′ − Θ ∗ −∪ i =1 N [ θ i ]is formed by the edges f = ( − e − e + e , − e + e ), f = ( − e + e − e , − e +2 e − e ), f = ( − e + e , e − e + e ), f = (2 e − e + e , e + e ), f =(2 e + e − e , e + 2 e − e ), f = (2 e − e , e + 2 e − e ). We first rule out thecase of the edges ( θ , θ − e ) and ( θ , θ + e ) being in S (or any other pair of edgesin the same relative geometrical positions as these two, with respect to Θ ′ ). Inthis case, f cannot be dominated by any copy of K : the two candidates, f − e and f + e cannot be in S . Because of this, three cases can be distinguished hereup to symmetry, for the 4-cycles corresponding respectively to the four cornersabove, namely: (a) (Figure 7) Θ = ( θ , θ − e , θ − e − e , θ − e ), Θ = ( θ , θ + e , θ + e − e , θ − e ), Θ = ( θ , θ − e , θ − e − e , θ − e ), Θ = ( θ , θ + e , θ + e − e , θ − e ). Then the following edges must be in S , dominating forcedly the edgesof F : f + e , f − e , f − e , f + e , f − e , f + e . The following 4-cycles areinduced by S : Θ = ( − e + e − e , − e + 2 e − e , − e + 2 e − e , − e + e − e )and Θ = (2 e + e − e , e + 2 e − e , e + 2 e − e , e + e − e ). The graphs12 igure 8. Four corners: instance (A), case (b ), subcase (b ) igure 9. Four corners, instance (A), case (b ), subcase (b ) ′ − Θ ∗ and Θ ′ − Θ ∗ have the respective vertices x = − e and y = e − e as non-corner vertices, so they cannot dominate z = − e and w = e − e ,yielding a contradiction. (b) Θ = ( θ , θ − e , θ − e − e , θ − e ), Θ = ( θ , θ − e , θ − e − e , θ − e ),Θ = ( θ , θ + e , θ + e + e , θ + e ), Θ = ( θ , θ + e , θ + e + e , θ + e ). Thenthe following edges must be in S , dominating forcedly the edges of F : f − e , f − e , f + e , f + e and possibly: (b ) (Figures 8-9) f + e , in which case: (b ) either Θ = Θ − e is in S and dominates Θ − e , so that x = − e + e − e cannot be dominated byany of its neighbors; (b ) or Θ = ( − e , e − e , e − e , − e ) is in S , so theend-vertices of the edge g = ( e − e , e + e − e ) cannot be dominated by S ; (b ) (Figure 10) f − e , in which case the end vertices of the edge g =( e − e , e + e − e ) cannot be in S or dominated by S , since h = ( e − e , e + e − e ) cannot be in S . (c) (Figure 11) Θ = ( θ , θ − e , θ − e − e , θ − e ), Θ = ( θ , θ − e , θ − e − e , θ − e ), Θ = ( θ , θ − e , θ − e − e , θ − e ), Θ = ( θ , θ + e , θ + e + e , θ + e ). Then the following edges must be in S , dominating forcedly theedges of F : f + e , f − e , f − e − f + e , f + e , f + e . It follows that x = − e cannot be dominated by S .Or Instance (B) : For the rest, we need by symmetry only to consider thecase in which the four corners of S in Θ ′ − Θ ∗ are θ = − e − e + e , θ =2 e − e + e , θ = − e + 2 e + e and θ = 2 e + 2 e + e . In the intersection ofthe affine plane < e , e > − e and Θ ′ − Θ ∗ , a 1-factor F is formed by the edges ofthe copies of K that should be dominated externally (off Θ ′ ) by induced copiesof K in S (parts themselves of 4-cycles induced by S ). We may assume that this1-factor is formed by the edges f = ( − e − e , e − e − e ), f = (2 e − e , e +2 e − e ), f = ( − e − e − e , − e − e ), f = ( − e + e − e , − e + 2 e − e ), f = (2 e − e − e , e − e ) and f = (2 e + e − e , e + 2 e − e ). It is enoughto consider by symmetry three cases of how F could be dominated externally, asjust mentioned, These cases have in common that f is dominated by f − e , f by f − e , f by f − e , and differ in that: (a) (Figure 12) f is dominated by f + e , f by f + e , f by f − e ; (b) (Figure 13) f is dominated by f − e , f by f − e , f by f + e ; (c) (Figure 14) f is dominated by f + e , f by f − e , f by f + e .In either case, by considering the dominating 4-cycle Θ = ( − e − e − e , − e − e , − e − e , − e − e − e ), the corresponding Θ ′ − Θ ∗ contains twocorners at distance 5, namely x = − e − e and y = − e + e − e , which wasruled out above.We just finished showing that there do not exist non-lattice like PDS[ Q ] sin Λ . Thus, the only standing case of a PDS[ Q ] in Λ is the lattice-like onethat remained by means of the commented programming code at the beginning15 igure 10. Four corners, instance (A), case (b ) igure 11. Four corners, instance (A), case (c)
17f the present proof that leads to the generator matrix (1) or its associated groupepimorphism Φ : Z → Z . This establishes the statement of the theorem. References [1] C. A. Araujo and I. J. Dejter,
Lattice-Like Total Perfect Codes , DiscussionesMathematicae Graph Theory, A Generalization of Lee Codes ,Designs, Codes and Cryptography, (2014), 77–90.[3] D. W. Bange, A. E. Barkauskas and P. J. Slater, Efficient Dominating Setsin Graphs , Appl. Discrete Math., eds. R. D. Ringeisen and F. S. Roberts,SIAM, Philadelphia, 1988, 189–199.[4] N. Biggs,
Perfect Codes in Graphs , J. Combin. Theory Ser. B, (1973),289–296.[5] J. Borges and I. J. Dejter, On Perfect Dominating Sets in Hypercubes andTheir Complements , JCMCC, (1996), 161–173.[6] S. Buzaglo and T. Etzion, Tilings by (0 . , n ) -Crosses and Perfect Codes. ,SIAM Jour. Discrete Math., (2013), 1067–1081.[7] I. J. Dejter, Perfect Domination in Regular Grid Graphs , Australasian Jour-nal of Combinatorics, (2008), 99–114.[8] I. J. Dejter and A. A. Delgado, Perfect Domination in Rectangular GridGraphs , JCMCC, (2009), 177–196.[9] I. J. Dejter, L. R. Fuentes and C. A. Araujo, There is but one PDS in Z inducing just square components ,arXiv:1706.08165.[10] I. J. Dejter and J. Pujol, Perfect Domination and Symmetry in Hypercubes ,Congr. Num., (1995), 18–32.[11] I. J. Dejter and O. Serra,
Efficient Dominating Sets in Cayley Graphs , Dis-crete Appl. Math., (2003), 319–328.[12] I. J. Dejter and P. Weichsel,
Twisted Perfect Dominating Subgraphs of Hy-percubes , Congr. Num., (1993), 67–78.[13] P. Dorbec and M. Mollard, Perfect Codes in Cartesian Products of 2-Pathsand Infinite Paths , Electr. J. Comb, (2005), p. ♯ R65.18 igure 12. Four corners, instance (B), case (a) igure 13. Four corners, instance (B), case (b) igure 14. Four corners, instance (B), case (c) Product Constructions for Perfect Lee Codes , IEEE Transactionsin Information Theory, (2011), 7473–7481.[15] T. Etzion, Tilings with Generalized Lee Spheres , in: J.-S. No et al., eds.,Mathematical Properties of Sequences and other Combinatorial Structures,Springer (2003), 181–198.[16] M. R. Fellows and M. N. Hoover,
Perfect Domination , Australasian Journalof Combinatorics, (1991), 141–150.[17] L. R. Fuentes, Perfect Domination and Cube-Sphere Tilings of Z n , Ph.D.thesis, University of Puerto Rico, Rio Piedras, June 2015.[18] H. Gavlas and K. Schultz, Efficient Open Domination in Graphs , ElectronicNotes in Discrete Mathematics, (2002), 681–691.[19] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domina-tion in Graphs, M. Dekker Inc., 1998.[20] P. Horak and B. F. AlBdaiwi, Diameter Perfect Lee Codes , IEEE Transac-tions in Information Theory, (2012),5490–5499.[21] W. F. Klostermeyer, A Taxonomy of Perfect Domination , J. Discrete Math-ematical Sciences and Cryptography, (2015), 105–116.[22] W. F. Klostermeyer and J. L. Goldwasser, Total Perfect Codes in GridGraphs , Bull. Inst. Comb. Appl., (2006) 61–68.[23] M. Livingston and Q. F. Stout, Perfect Dominating Sets , Congr. Numer., (1990), 187–203.[24] P. M. Weichsel, Dominating Sets in n-Cubes , J. Graph Theory,18