In Congressus Numerantium 97 (1993), pp. 51–70. Perfect Dominating Sets on Cube-Connected Cycles Douglas Van Wieren Marilynn Livingston Quentin F. Stout Advanced Computer Architecture Laboratory Electrical Engineering and Computer Science University of Michigan, Ann Arbor Abstract Cube-connected cycles are a family of cubic graphs with relatively small diameters and regular structure, making them attractive models for parallel architecture design. The existence of perfect dominating sets for any structural model of parallel computation is both useful for the construction of efficient algorithms for that structure and indicative of practical design constraints. This paper gives a simple algorithmic method for constructing perfect dominating sets on cube-connected cycles where they exist, and proves nonexistence for all other cases. Specifically, standard perfect dominating sets (distance equal to 1) are shown to exist for cube-connected cycles of order k, k not equal to 5. Moreover, the existence of perfect dominating sets for all distances greater than 1 is disproved (with the trivial exception — the distance equaling or exceeding the diameter of the graph). Keywords: Cube-Connected Cycles, Dominating Sets, Perfect Dominating Sets, Parallel Architecture, Parallel Algorithms. 1 Notation and Background 1.1 Cube-connected cycles Formally, a cube-connected cycle of order k 1 (here denoted CCCk ) can be described as the labeled graph (V; E ) where V , the vertices of CCCk , is the set n i 2 f0 n g 2 , the edges of CCC , is the set of unordered pairs f( (i; j ) : ;:::;k 1 +1  i2 0; : : : ; 2 i1 ; j1 ); and E k elements of V which satisfy either i1 ;j (mod k) and or i1 = i2 and j1 j j1 k 1 oo g where ( (i2 ; j2 ) = i1 ; j1 ) and (i2 ; j2 ) are j2 j2 j=2 (k i1 1) : The edges which satisfy the first condition are referred to as cycle edges; the remaining edges, exactly those which satisfy the second condition (i1 = i2 and jj1 j2 j = 2(k i1 1) ), are referred to hypercube edges. 1 Cube-connected cycles of orders smaller than 3 are traditionally ignored in the same manner that cycles of order smaller than 3 are ignored. However, various reasonable extensions of the definition (i.e. nonsimple graphs) will produce graphs small enough to allow most results (including this one) to shown by inspection. 1 By inspection, the removal of all hypercube edges produces a graph with 2k components, each of which is a k -cycle. Thus, contracting all the cycle edges in CCCk will produce a graph with 2k vertices (in fact, a hypercube). For this reason, each k -cycle in CCCk which does not include any hypercube edges is referred as a supervertex of the (embedded) hypercube. Moreover, the origin of CCCk is the set of vertices in the supervertex located at 0. In essence, the formal definition uniquely describes each vertex in CCCk by its position within the supervertex (the cycle index) and by the position of the the supervertex within the hypercube. Following the conventions established for the hypercube, the position of a supervertex will be described as a binary string (e.g. 010000000 or, in the form of a regular expression, 010(7) ). Specific vertices will be described with the addition of an accent over the bit position corresponding to the index. (For simplicity, the bit positions will indexed from 0 to k - 1 from left to right.) An example of a cube-connected cycle and the notation used here is shown in figure 1. Informally, two vertices are adjacent if either they are located in the same supervertex and the corresponding indices are adjacent in the k -cycle or they each have the same cycle index and have supervertex locations which differ only in the bit position indicated by the index (accent). It follows that, for any k , each vertex in CCCk has degree 3, and, with this notation, the three neighbors can be described simply — e.g., in CCC6 , 1010^11 has neighbors, 101^011, 10101^1 , and 1010^01. ^ 111 ^ 101 ^ 101 ^ 101 ^ 110 ^ 100 ^ 110 ^ 100 ^ 100 ^ 110 ^ 001 ^ 001 ^ 011 ^ 001 ^ 011 ^ 000 ^ 000 ^ 111 ^ 111 ^ 011 ^ 010 ^ 000 ^ 010 ^ 010 Figure 1: Cube-Connected Cycle of order 3 From the informal definition, it should be intuitive that, for any vertices v0 and v1 in CCCk , there exists a graph isomorphism which sends v0 to v1 — rotation or reversal of the dimensions (and corresponding cycle indices) and/or reflection over any dimension as needed. A perfect dominating set of distance d (here abbreviated PDSd ) is a subset of the vertices of graph such 2 that every vertex in the graph is dominated by exactly one vertex in the set. Formally, for any graph G, let d(v; w) be the minimal path distance between the vertices v and w (with d(v; v ) = 0). For a given positive integer distance d, a vertex v is said to cover or dominate w if and only if d(v; w)  d. Letting V (G) denote the set of vertices of G, a subset X of V (G) is a dominating set with distance d if and only if 8v 2 V (G); 9w 2 X such that w covers v. A subset X of V (G) is a perfect dominating set with distance d if and only if 8v 2 V (G); 9 a unique w 2 X such that w covers v . A standard perfect dominating set 2 is a perfect dominating set of distance 1; an example of a standard perfect set is shown in table 1 and figure refpds3. 1.2 Motivation and Background Cube-connected cycles, as mathematical structures, are interesting in and of themselves; however, they have two properties which make them particularly attractive as potential structures for massively parallel computers. First, like the mesh, but unlike the hypercube, each node has a small, fixed degree. This allows the redesign of larger systems without the need to redesign and rethink the individual processors. Also, from the viewpoint of theoretical computer science, asymptotic analysis of hypercube algorithms holds troubling questions regarding the computational power of processors which have  (log n) connections. Second, like the hypercube, but unlike the mesh, the diameter of the graph grows slowly with respect to the p number of processors ( (log n) as opposed to  ( n)). Since algorithms designed for parallel architectures often require data from all processors, reduction of the worst-case communication time may be a matter of necessity. Dominating sets are also an area of strong concern in the design of both parallel structures and parallel algorithms. With a specified processor structure (such as the mesh, hypercube, or cube-connected cycle), it is often necessary to find an efficient method of distributing limited or costly replicable items — power sources, i/o ports, function libraries, algorithm information, etc. — among the processors. In some variations of the problem, resources may conflict, and, in fact, with regular structures, having resources placed within some (short) distance of every node is not always sufficient. Other considerations – such as the complexity of the paths between each processor and its designated resource – are also considerations. Because of regularity of the structures, a perfect dominating set is usually the best answer. The existence of perfect dominating sets for the mesh family of architectures is fairly straightforward (depending on the mesh) and the hypercube has been investigated exhaustively for this property however, results for the cube-connected cycle architecture were not generally known beyond k > 12 and d > 1. Here, an algorithmic method is shown for constructing a PDSd on CCCk when d = 1 and k 6= 5. The nonexistence of nontrivial perfect dominating sets when d > 1 is also demonstrated. 2 Algorithmic Construction of Standard Perfect Dominating Sets The existence of perfect dominating sets for d = 1 and k 6= 5 will be shown by explicit construction. In addition to describing the vertices which belong to this set, it will also be useful to distinguish between the non-member vertices by the direction in which the vertex which dominates it lies — not only will this make the allocation scheme adaptable to a wide variety of uses, it will also be used to demonstrate correctness. 2 The term standard dominating set is intended to agree with various definitions of dominating set used when distance is not a consideration. 3 Location of Supervertex 000 001 010 011 100 101 110 111 Table 1: Marking of CCC3 Index 0 CH CR R CL R CL CR CH CH CL CR R CR R CL CH Index 2 CH R CL CR CL CR R CH showing a Standard Perfect Dominating Set 10 00 01 10 11 Index 1 11 01 CH CH CH CR CL R R CR CL CL R CR Table 2: First Labeling Scheme Component 2.1 Vertex Marking Each vertex in CCCk has two neighbors within the cycle and one neighbor adjacent along its hypercube edge. Thus, four possibilities exist for each vertex. Either it is a resource node (in the perfect dominating set), the vertex which covers it lies across the hypercube edge, or the vertex which cover it lies to the left or right within the cycle. Formalizing the description: R a member of the perfect dominating set (a Resource node) CH dominated by the adjacent vertex along the dimension indicated by the index (Covered CL CR along a Hypercube edge) dominated by the adjacent vertex within the same cycle to the left — meaning the vertex whose index is 1 more ( modk ) (Covered by a node to the Left) dominated by the adjacent vertex within the same cycle to the right — meaning the vertex whose index is 1 less ( modk ) (Covered by a node to the Right) A solution for k = 3 (d = 1) is presented with these labels in table 1 (inspection of figure 2 suffices to verify that this is, indeed, a PDS1 ). 2.2 The Method The general solution for d = 1, k = 5 is generated using copies of Table 2 and Table 3. First, the value of k is decomposed into 3a + 4b where a and b are nonnegative integers (note that this is possible for all k > 3 with the exception of 5). Then, a copies of the first component (table 2) are concatenated with b copies of the second component (table 2). As an example, when k = 11, the example shown in table 4 might be produced. 6 4 ^ 101 ^ 111 R CH CR CL ^ 101 CH ^ 101 CH ^ 111 ^ 111 ^ 110 ^ 100 CR CL R CR ^ 100 CL ^ 110 ^ 100 CL R ^ 110 ^ 011 ^ 001 CR R CR CL R ^ 001 ^ 001 ^ 011 ^ 011 ^ 010 ^ 000 R CH CH CL ^ 000 ^ 000 ^ 010 CR ^ 010 CH Figure 2: Cube-Connected Cycle of order 3 with Perfect Dominating Set (d = 1) 10 10 10 10 CH CL R CR CR CH CL R R CR CH CL CL R CR CH 00 01 10 11 Table 3: Second Labeling Scheme Component Table 4: Example Concatenation of Components 10 00 01 10 11 11 01 10 10 10 10 10 10 10 10 CH CH CH CH CL R CR CH CL R CR CR CL R CR CH CL R CR CH CL R R CR CL R CR CH CL R CR CH CL CL R CR CL R CR CH CL R CR CH 5 Now, given the position of any vertex in CCCk , its index and the the location of the supervertex within the hypercube, the composite table can be used to provide a marking for each vertex such that the vertices marked with R form a perfect dominating set. The binary values immediately above the table (in our example, 10, 11, 01, 10, 10, 10, 10, 10, 10, 10, 10) for which the corresponding bit is high are bitwise XOR’ed together (using 00 as the result for the supervertex at the origin). The resulting binary number is the label of the row which will be used. The column is that of the index. Thus, in order to find the appropriate designation for 10^100100000, note that the supervertex located at 10100100000 has high bits in three locations — the first, third and sixth positions from left to right — corresponding to values 10, 01, and 10; since 10  01  10 is 01, the row labelled 01 will be used; the column is that underneath the accented 1; and, hence, 10^100100000 will be marked as a resource node. 2.3 Correctness: The row chosen is dependent only on the location of the supervertex. Thus, the labeling within each cycle can be read directly from the appropriate row of the table. Hence, it can quickly verified that, from left to right (and wrapping around the end), the constraints on a PDS1 are followed within each supervertex — that each vertex with designation R has left and right neighbors appropriately labeled CR and CL, etc. (note that the tables above are constructed so that concatenation preserves this property). What may or may not be so quickly verified is that edges between supervertices are also properly used (a vertex is marked with CH /R if and only if it has a neighbor along its hypercube edge with designation R/CH ). Note that two supervertices connected to one another along dimension i are connected at the vertex with index i. Thus, two vertices which are connected by an edge not in a cycle (a hypercube edge) have the same index, i; the designations, therefore, come from the same column. Note also that supervertex locations must differ in exactly one bit — the bit with index i. Thus, the row used to label one vertex is different from that vertex’s (only) cube-edge neighbor by an XOR operation with entry above column i. Now, a closer examination of the columns of each initial table confirms that, for any row label, bitwise XOR’ing with that binary value will not map the designation R/CH onto any designation than CH /R. Now, since every vertex has a designation, R, CH , CL, or CR; no vertex with designation R has any neighbor marked incorrectly, and each of the markings CH , CL, and CR correctly indicate the presence of a resource node, it follows that the process of marking indicated by the tables has, in fact, generated a perfect dominating set. 2.4 A Proof of the Non-existence of a PDS1 for k = 5 It does not follow that no PDS1 exists when k = 5 simply because the above generating scheme does not produce a proper marking for that case. The argument for nonexistence is more subtle. There are 5  25 (160) vertices in CCC5 . If a PDS1 did exist, since each vertex covers exactly four neighbors, it would contain exactly 5  25 2 (40) elements. By inspection, no supervertex can contain more than one element from the PDS1 . Since there 25 (32) supervertices, by the pigeon-hole principle, some supervertex must contain more than one element. Hence, the assumption that a PDS1 did exist is contradicted. 6 3 Nonexistence of Perfect Dominating Sets with Greater Distances With the exception of the distance equaling or exceeding the diameter of CCCk , when no PDSd for CCCk exists when d > 2. This can be restated as no PDSd with more than one vertex exists for CCCk when d > 2. (Note: If d is less than the diameter of CCCk , more than one vertex is necessary — this is not true for all graphs, but follows from the automorphisms of CCCk .) The argument for nonexistence proceeds as follows: When d  5, the pattern of vertices covered by a single vertex is sufficiently “irregular” to preclude the construction of a PDSd with more than one element. For 2  d  4, examination of various conditions on the number of vertices covered by a single vertex will handle the majority of the remaining cases. 3.1 Elimination of Large Values of d The first set of cases will be all but a finite number of values for d. Specifically, the existence of a PDSd with more than element is shown not to be possible when d  5. The general technique is to choose an arbitrary vertex v0 from CCCk , and demonstrate that there exists a vertex v which is isolated — it is not covered by v0 and any vertex which does cover it also covers some vertex which is already covered by v0 . Thus, if we start to construct a perfect dominating set by the inclusion of v0 , we cannot include a vertex which covers v without having some vertex covered twice. (By the symmetries of CCCk , any arbitrary vertex is isomorphic to a particular vertex; so, it is sufficient to demonstrate an isolated vertex v when v0 is ^00k 1 . This vertex will be referred to as the zero of CCCk .) 3.1.1 Useful Definitions and Lemmas Lemma 3.1.1 Any path in CCCk with the locations of the supervertices containing the endpoints differing in bit position p must include a visit to a vertex with index p. P ROOF : The set of hypercube edges along dimension i form a cutset; thus, the described path must include one of these edges. The lemma follows immediately from the fact that the only vertices incident to these edges have index i. Lemma 3.1.2 (The Distance Calculation Lemma) Let v0 and v1 be vertices in CCCk such that v0 has index i0 , v1 has index i1 , and the locations of the supervertices containing v0 and v1 differ in the bit positions described by the set H fp0 ; : : : ; ph 1 g. If x be the number of edges in the shortest walk on the k-cycle starting at index i0 and ending at index i1 which includes a visit to every vertex with an index in the set H then x h is the distance in CCCk from a vertex v0 to a vertex v1 . = + P ROOF : Without loss of generality, let v0 be vertex i0 at the origin in CCCk , and let v1 be vertex i1 in the supervertex whose location has high bits in positions described by H . Any path from v0 to v1 must include at least h hypercube edges. Removing these edges from the path and mapping the remaining cycle-edges onto the k -cycle in obvious manner will form a walk on the k -cycle. This walk starts at vertex i0 , ends at vertex i1 , and visits every vertex with an index in the set H . The length of this walk must be at least x. So the distance from v0 to v1 is least x + h. Now, given a walk W in the k -cycle from vertex i0 to vertex i1 which visits every vertex with an index in the set H which has length x, we can construct a walk from v0 to v1 in the following way: Starting at v0 , if the 7 current vertex is the first vertex encountered with an index in H , include the hypercube edge and proceed; otherwise take the next edge in W , map it onto the current supervertex, and proceed along that edge. This walk has length x + h, so the distance from v0 to v1 must be at most x + h, and the lemma follows. Corollary 3.1.1 Let v0 ; v1 ; v2 be vertices in CCCk . The distance from v0 to v1 is less than the distance from v0 to v2 if the index of v1 is equal that of v2 and set of bit positions where the supervertices containing v0 and v1 differ is a proper subset of the set of bit positions where the supervertices containing v0 and v2 differ. Lemma 3.1.3 The diameter (the greatest distance between any two vertices in a specified graph) of CCCk 2 when is 52k 3; when = 3, the diameter is 6. b k> k P ROOF : Let v0 be the zero of CCCk (equivalent to an arbitrary choice). Now, by the above corollary, if v1 is any vertex in CCCk with index i1 , then the distance from v0 to v1 is not greater than the distance from v0 to the vertex with index i1 in the supervertex whose location has only high bits. Thus, it is sufficient to find the index i1 such that the shortest walk from 0 to i1 on the k -cycle which visits all vertices is maximal. By inspection, i1 = 0 suffices for the case k = 3 and i1 = b k2 suffices for all other cases. Lemma 3.1.4 CCCk . k > 45d is a necessary condition for the existence of a PDSd with more than one vertex for P ROOF : If a PDSd with more than one element exists for CCCk , then the shortest path between any two elements in that PDSd is at least 2d + 1. Because that distance is at most the diameter of CCCk , when 5k 5k 2 , and immediately d < 4 . When k = 3, the diameter is 6, and the maximum k  4, 2d + 1  b 2 allowable distance for a PDSd is 2; thus, the lemma follows. 3.1.2 Isolated Points for Certain Values of d Let d = 8, and, therefore, k > 6, and the vertices v n1 n2 n3 = = = = (k ^ 1010 (k 1010 (k ^ 0010 (k ^ 1010 4) 1 4) ^ 1 4) 1 4) 1 are well-defined. Now, v is not covered by (within distance d of) the zero of CCCk . This is the first application of the distance calculation lemma; so, explicitly: the shortest walk on the k -cycle from 0 to 0 which visits vertices in f0; 1; 2; k 1g has length 6 (by inspection — recall that k is least 6, making a cycle tour at least as long as the walk given by the vertex sequence (0; 1; 2; 1; 0; k 1; 0)); thus, the distance from the zero of CCCk to v is 6 + 3 (the number of hypercube edges traversed and/or the number of high bits in the location of the supervertex containing v ). However, all of the neighbors of v (exactly n1 , n2 , and n3 ) are covered by the zero of CCCk (this also follows from the distance calculation lemma). Since any vertex which covers v must also cover at least one of its neighbors, the vertex v is isolated. (In fact, this is a stronger condition – for every vertex v1 such that d(v; v1 ) = d, any path of length d connecting v1 and v must include a vertex which is covered by the zero.) 8 Thus, it is not possible, when d = 8, to select a set of vertices which cover all vertices in CCCk without some vertex being covered at least twice. Hence, when d = 8, no PDSd exists for CCCk . Similarly, when d = 10 and v = ^1010(k 6) 10, or d = 13 and v = ^10110(k 7) 10, v can be shown to be isolated. In both cases, the zero of CCCk does not cover v , but does cover all three neighbors of v . Not every value of d where d  5 will produce a vertex which is not covered by the zero, but has all three neighbors covered by the zero. However, it will be possible to demonstrate a vertex v which is not covered by the zero and a set B of vertices which are covered by the zero so that every vertex which covers v must cover some vertex in B . = 5, ^ (k = 010 For d let v = ^ 010(k 3) 1, and = S B f g are well defined. b2 b3 and let B be the set fb0 ; b1 ; b2 ; b3 g where b0 = 1^00(k 3) 1, b1 = 110(k 3) ^0, 010(k 3) ^ 1. Every vertex in B is covered by the zero, v is not, and all vertices in 3)1 v Let v1 be any vertex which covers v . If d(v; v1 ) < 5, v1 must cover b2 and b3 as well as v . Otherwise, d(v; v1 ) = 5 and, if v1 does not cover either b2 and b3 , d(^110(k 3) 1; v1 ) = 4. Moreover, either d(1^ 10(k 3) 1; v1 ) = 3 or d(110(k 3) ^ 1; v1 ) = 3. Assume that d(110(k 3) ^ 1; v1 ) = 3; since d(110(k 3) ^ 1; b1 ) = ( k 3) ( k 3) 1, d(b1 ; v1 ) = 4. Otherwise, d(1^ 10 1; v1 ) = 3; since d(1^ 10 1; b0 ) = 1, d(B1 ; v1 ) = 4. Thus, any vertex which covers v must cover some vertex in B and v is isolated. A similar argument suffices for the case d = 7 with v Thus, when CCCk . 3.1.3 d (k =^ 0010 4) 1 and B= n (k 00^ 10 4) (k 1; 00^ 00 4) 1; 0010 (k 4) ^ 1; 0010 (k o 4) ^ 0 : 2 f5 7 8 10 13g , an isolated vertex exists, and it is not possible to construct a PDSd ; ; ; ; for Isolated Points for Large Values of d Let d = 3n+2m+i where n, m, and i are integer values such that n  m  i > 0. Note that appropriate values for n, m, and i can be chosen to produce any positive integer except those in f1; 2; 3; 4; 5; 7; 8; 10; 13g: Let v denote ^11(n) 0(k m n 1) (m) 1 . Since k > 4(3n + 2m + i) 5 +i m > 2n + m + > 2n + m + i > n 5 +m+1 this v is well-defined. Now, the length of the shortest walk on the k -cycle from 0 to 0 which visits all vertices in = fm k; m k + 1; : : : ; k 1; 0; 1; : : : ; ng is the minimum of k (corresponding to a complete cycle tour) and 2n + 2m (any other walk must visit both n and k m and, since it is closed, each edge will be included twice). Since k > 2n + m + i and m  i implies that 2n + 2m  2n + m + i, 2n + m = i is a lower bound on the length of that walk. Thus, by the distance calculation lemma, the distance from the zero of CCCk to v is at least (2n + m + i) + (n + m + 1), which is greater than d. Thus, v is not covered the zero of CCCk . H Let Bn be the set n B0 B1 : : : Bn ^ (k 1 Bn 0 m n 1) Bk m : : : Bk 9 1 : 8 2 j H; Bj 2 f0 1g ; o and let Bm be the set n and let B = Bn (k B0 B1 : : : Bn 0 SB m m n 1) Bk^ m Bn k +1 : : : Bk 1 : 8 2 j H; Bj 2 f0 1g ; o . Both 1(n) ^10(k m n 1) 1(m) and 1(n+1) 0(k m n 1) ^11(m 1) are covered by the zero of CCCk (the length of the shortest walk from 0 to n in the k -cycle which includes vertices from H is at most n + 2m; the length of the shortest walk from 0 to k m in the k -cycle which includes vertices from H is at most 2n + m; (n + 2m) + (n + m + 1)  (2n + m) + (n + m + 1)  d). Thus, by the corollary to the distance calculation lemma, it follows that every element of B is also covered by the zero of CCCk . Let v1 be any vertex in CCCk within d distance of v . Let i1 denote the cycle index of v1 , and let H1 be the set of bit positions where supervertices containing v and v1 differ. Now, choose a vertex bn from Bn and a vertex bm from Bm such that 8j 2 H , the j th bit of the location of the supervertex containing bn and bm agrees with the corresponding bit of the supervertex containing v1 . If H1  H (the set of indices from H1 which do not appear in H ) is empty, both bn and bm lie in the same supervertex as v1 and, immediately, at least one of bn and bm is within distance d of v1 (v1 is within distance d of the vertex with cycle index 0). Otherwise, the shortest walk on the k -cycle from i1 to 0 which visits all vertices from H1  H ends with a path from either n or k m to 0. Thus, there is a walk on the k -cycle starting at i1 , visiting every vertex in H1  H , and ending at either n or k m which is shorter. Since H1  H  H1 , one of either bn or bm is closer to v1 than v . Since any vertex which covers v also covers some vertex already covered by the zero of CCCk , if d is any value not in f1; 2; 3; 4; 5; 7; 8; 10; 13g  f5; 7; 8; 10; 13g , no PDSd exists containing more than one element for CCCk . C OMMENT: When d < 5, it is possible to show that no isolated vertices exist. 3.2 Nonexistence of Perfect Dominating Sets for 2  d  4 Let C (d; k ) be the number of vertices in CCCk covered by a single vertex. By the isomorphic properties of CCCk , C (d; k ) is a constant for fixed k and d. In fact, it will be shown that when k > 2d, C (d; k ) depends only on the value of d. For most of the remaining cases, when 2  d  4, C (d; k ) will shown later to contain an odd prime factor p which either does not divide k or is greater than 2d. Both conditions preclude the existence of a PDSd for CCCk . The first is shown immediately with the following lemma: Lemma 3.2.1 If there exists a PDSd for CCCk and divisible by p. C (d; k ) contains an odd prime factor p, then k is P ROOF : There are k  2k vertices in CCCk . Each element in the PDSd must cover exactly C (d; k ) vertices; 2k so, Ck(d;k (the number of vertices in a PDSd ) is an integer, and the lemma follows. ) The structure of the argument for the second will be to show that a necessary condition for the existence of a PDSd for CCCk when k is a multiple of exactly n factors of an odd prime p such that p > 2d is that the number of elements in the PDSd must be divisible by pn (this is trivially true when n = 0). Therefore, k for any prime p > 2d, given that k contains exactly n factors of p, pkn C (2d;k) must be an integer — implying immediately that C (d; k ) cannot contain any prime factor greater than 2d. Hence, if C (d; k ) does contain a prime factor p greater than 2d, no PDSd exists for CCCk . 10 3.2.1 Necessary Definitions By the automorphisms of CCCk , if some vertex with index i covers exactly n vertices with index j , then every vertex with index i covers exactly n vertices with index j . The following definitions make that property explicit and offer a straightforward method of calculating the value of n. Definition 3.2.1 Let C (d; k)j be the number of distinct subsets H of 0; : : : ; k 1 such that there exists a walk of length x on the k-cycle which starts at index d, visits all vertices in H , and ends at index j subject to x + H d. f g j j Lemma 3.2.2 C (d; k)j is the number of distinct vertices with index tance of a single vertex with an index d. j in CCCk which are within d dis- P ROOF : This follows from the distance calculation lemma. With H taken as a set of bit positions, each distinct subset of H uniquely describes a supervertex location within CCCk and, together with j , a unique vertex in CCCk . Corollary 3.2.1 The number of vertices covered by a single vertex, C (d; k ), is X k 1 i=0 C (d; k )i . Corollary 3.2.2 For all i; j 2 f0; : : : ; k 1g, the number of distinct vertices with index covered by a single vertex with index j is C (d; k )(d+i j mod k) . i which can be C LARIFICATION : This follows from the automorphisms of CCCk . If a vertex with index j covers vertices with index i, then a vertex with index (j + x mod k ) covers vertices with index (i + x mod k ). It now follows that, for large values of k , C (d; k ) does not depend on k . Since our concern is now with a finite set of values of d, the set of values of C (d; k ) and C (d; k )j under consideration will also be finite (and calculable by the method implicit in the definition of C (d; k )j ). Lemma 3.2.3 For all k > 2d, C (d; k)j = C (d; 2d + 1)j . P ROOF : When k > 2d, the value of C (d; k )j does not depend on k . No walk of length at most d starting at d visits any vertex outside of the set S = f0; 1; : : : ; 2d 1; 2dg. Immediately, any H 6 S need not be considered and 8j 62 S , C (d; k )j = 0. Corollary 3.2.3 For all k > 2d, C (d; k ) Corollary 3.2.4 For all j > 2d, C (d; = C (d; 2d + 1). 2d + 1)j = 0. Now, given that a single vertex with index j covers exactly C (d; k )(d+i j mod k) vertices with index i, if Xj is the number of elements in a PDSd for CCCk with index j , then the sum over j of Xj C (d; k )(d+i j mod k) is total number of vertices in CCCk with index i. This property and similar necessary conditions can be expressed as matrix relationships. The following definition will be useful in that regard. 11 0 8 10 10 B B 11 B 6 B 1 B 0 B 0 B 0 B B 0 B 0 B 0 B 0 B 0 B 0 B B 0 B 1 B  6 11 10 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 1 6 11 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 1 6 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 1 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 Figure 3: Example: Definition 3.2.2 For all C (d; 2d + 1)(d+i n 2 d + 1, let M (d; n) 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 1 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 6 1 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 11 6 1 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 10 11 6 1 0 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 1 CC CC CC CC CC CC CC CC CC A M (5; 21) denote the j mod n) . 0 0 0 0 0 0 0 0 0 1 6 11 10 10 8 10 10 11 6 1 0 n  n matrix such that M (d; n)i;j = An example of such a matrix is shown in figure 3. 3.2.2 Properties of M (d; pn ) Examining the definition, we find that M (d; n)i;j = M (d; n)(i+1 mod n);(j +1 mod n) . In essence, each row of M (d; n) is the previous row “shifted” right (with wrap-around). Other properties of interest are: Lemma 3.2.4 Each row distinct nonzero entries. R in M (d; n) is nonzero, has no negative entries, and, in fact, has at least two P ROOF : Applying the calculation method implicit in the definition of C (d; 2d + 1), C (d; 1) = d + 1. Let R be the j th row of M (d; n), and, WLOG, assume j  d, the (j M (d; n)(j d);j ) , is 1 and the (j d + 1)th entry in R is d + 1. C (d; 0) = 1 and d)th entry in R, Lemma 3.2.5 When n = qp where p > 2d, each row R in M (d; n) has the property that for all j , the ith entry in R is nonzero for at most one value of i j (mod p).  P ROOF : Recalling that C (d; 2d + 1)j = 0 for all j  2d + 1, then C (d; 2d + 1)(d+i j mod n) nonzero implies that d + i j mod n is less than 2d + 1 and, therefore, less than p. If i0  i1 (mod p), but i0 6 i1 (mod p)q , then at most one of d + i0 j mod pq and d + i1 j mod pq is less than p. By the definition, at most one of M (d; n)i0 ;j and M (d; n)i1 ;j is nonzero. Lemma 3.2.6 M (d; n) has rank at least n 2d. P ROOF : By inspection, the rows d through n d 1 are in diagonal form. (When j 2 fd; d + 1; : : : , n d 1g and i < j d, d + i j is an element of f2d + 1 n; 2d + 2 n; : : : ; 1g. Since n  2d + 1, (d + i j mod n)  2d + 1 and M (d; n)i;j = 0.) 12 Theorem 3.2.1 Let n be a positive integer, M (d; pn ) is regular when p > 2d and p is prime. P ROOF : Let L be the isomorphism from (i + 1 mod pn )th standard basis vector. <p ( n) to itself which sends the ith standard basis vector 3 to the Treating L purely as a linear transformation, the minimum polynomial of L is () = (p ) 1. (The minimum polynomial of a linear transformation L is the nonzero polynomial  of least degree such that 8v; ((L))(v) = 0 4.) Verification is immediate since both L(pn) and Ip(0) are the identity transformation, thus n 8v 2 <p; ((Ip))(v) = (Ipp Ip(0) )(v) = Ip(p) (v) ( ) Ip(0) (v) = 0 Minimality follows from consideration of the operation of any nonzero polynomial of degree less than pn on a standard basis vector. Now, for any nonzero vector v , the order of v with respect to a linear transformation L is the nonzero polynomial v of least degree such that (v (L))(v ) = 0. The minimum polynomial  is a multiple of the n order of any vector; thus, for any nonzero vector v 2 <(p ) , the only possible values for v are factors of n n (p ) 1. As shown in appendix 3.4, a complete factorization of (p ) 1 is f 8 [ <pX ip 1g :i  1 ( j) : j 2 f0; 1; : : : ; n =0 n o L(n) (v) : n > 0 . (Formally, S (v) is an invariant subspace of <(pn) , the cyclic space relative to L generated by the vector v.) The degree of the order of v is the rank of the subspace S (v ). (Let v have degree r , v (L)(v ) = 0 implies that L(r) (v ) is a linear combination of Define S (v) as the subspace spanned by 9 = 1g : ; n vectors from n L(n) (v) : 0  n  r o Since v (L)(v ) has minimal degree, there does not exist r0 vectors from L(n) (v ) : 0  n  r0 1 . Thus, S (v ) has n Ip(n) (v) : 0  n  r 1 o :  r such that Lr (v) is a linear combination of 0 where r is the degree of v 1, o as a basis, and, directly, the rank of S (v ) is the number of vectors in that basis, r .) Now consider the order of R, an arbitrary row of M (d; pn ). S (R) is, by definition, the subspace of <(p ) spanned by the row vectors of M (d; pn ) (L(i) (R) is also a row in M (d; pn )) for arbitrary i; and for any row R0 in M (d; pn ), there exists an i such that R0 = L(i) (R)). Thus, degree of R is the rank of M (d; pn ). n Since R is nonzero and has no negative entries; (p 1 + p 2 +


+  + 1)(L)(R) is a nonzero vector n n and R is not a factor of p 1 + p 2 +


+  + 1. Thus, R must be a multiple of ( 1). n n Recall that R has two distinct nonzero entries and that for all j , the ith entry in R is nonzero for at most one value of i  j (mod p). Thus, 0p X R0 =  (n 1) i=0 1 1  ip A (L)(R) ( ) ~ 2 <m with the nth entry, the only nonzero entry, equal to 1. As The nth standard basis vector for <m is the unique vector V used here, the enumeration of positions begins with 0; thus, the 0th standard basis vector denotes (1; 0; : : :). 4 The definition often includes the restriction that the leading coefficient must be 1 to avoid ambiguity. 3 13 is a nonzero vector with at least two distinct entries. Immediately, ( nonzero. Since (( 0 1)(L))(R ) applied to that vector is 00 11 1 X (( 1)(L))  ( ) )(LAA (R) =0 0 0 11 1 X ( 1)  ( ) AA (L)(R) =0 ! ( ) 1 P 1  (L)(R) p = 1)(L) (n 1) ip i (n 1) p = ip i n p = P is a nonzero vector, R has ( p p i=0 1 i=0 i  ) as a factor. i Since the rank of M (d; pn ) is at least pn 2d,  cannot be a factor of R ( P p n) 1 i=0 p 1  (ipj ) for j  1 since, otherwise, the degree of R would be pn P j Thus, R must include ( pi=01 (ip ) ) as a factor for all j (p  0. 1)pj which, when j  1 is less than p n 2d. n It follows that the order of R must be (p ) 1. The degree of R being pn , the rank of M (d; pn ) is therefore pn and, since M (d; pn ) is a pn  pn matrix, M (d; pn ) is regular (invertible). 3.2.3 The Relation of M (d; k ) to a PDSd for CCCk Now the regularity of M (d; pn ) is applied to show the following theorem: Theorem 3.2.2 Let k = qpn where p is an odd prime such that p > 2d, and n and q are positive integers. If there exists a PDSd for CCCk , then the number of elements in the PDSd must be divisible by pn . P ROOF : Let k , p, n, and q be as described, and define the pn -index of a vertex as its cycle index mod pn (this notation is purely for convenience of expression). Now, the proof can be expressed in the following two lemmas. Lemma 3.2.7 M (d; pn )i;j is the number of distinct vertices in CCCk with pn -index j which can covered by a single vertex with pn -index i P ROOF : By an earlier lemma, the number of distinct vertices with index j which can be covered by a single vertex with index i is C (d; k )(d+i j mod pn ) ; therefore, for j 2 f0; : : : ; pn 1g, the number of vertices with an index equivalent to j modulo pn which can be covered by a single vertex with index P i (WLOG, pn-index i) is `q=01 C (d; k)(d+i+`pn j mod k) . Now, C (d; k)(d+i+`pn j mod k) is zero when d + i + `pn j mod  2d + 1, implying that, C (d; k)(d+i+`pn j mod k) is nonzero for at most value of P `. In fact, `q=01 C (d; k)(d+i+`pn j mod k) = C (d; k)(d+i j mod pn ) . Because M (d; pn )i;j = C (d; 2d + 1)(d+i j mod pn ) = C (d; k )(d+i j mod pn ) , the lemma follows. ~ denote the column vector in Definition 3.2.3 Let X be any set of vertices from CCCk and let X n ~ ith entry, Xi , is the number of elements in X with p -index i. 14 < k whose  ~ = 2k q 2k q Lemma 3.2.8 If X is a PDSd , then M (d; k)  X


T 2k q . P ROOF : If X is a PDSd , then every vertex in CCCk with pn -index j must be covered by exactly one vertex from X . The number of distinct vertices with pn -index j covered by a single vertex with pn -index n ~ . No vertex in CCC i is M (d; k )i;j ; the number of elements in X with p -index i is X is covered by i k ~ more one element of X ; so, M (d; k )i;j multiplied by Xi . must be the total number of distinct vertices with n n n p -index i covered by the vertices in X with p -index j . It follows that if Rj is the j th row of M (d; p ), ~ must be the total number of distinct vertices with pn -index j covered by the set X . Since X is a then Rj X PDSd , this be must the total number of vertices in CCCk with pn -index j . The lemma then follows from the definition of matrix multiplication and the fact that there are exactly 2k vertices in CCCk with any arbitrary index i — and, thus, 2k q vertices with any arbitrary pn -index i. ~ such that M (d; k )  Y ~ Since M (d; pn ) is regular, any pn length column vector Y  2k q 2k q =


2k q T ~ is must have uniform entries. (One solution for X P n 1 p i=0 2k q M (d; p n )i;j P n 1 p i=0 2k q M (d; p n )i;j


P p n 1 i=0 !T 2k q M (d; p n )i;j ~ must be Immediately, since M (2d; pn ) is invertible, this is the only solution.) Noting that the entries in X n ~ integers, it follows that the sum of the entries in X must be an integer which is divisible by p . Under the assumptions stated earlier, the number of vertices in X , an arbitrary PDSd for CCCk , is the sum of the ~ ; thus, the number of elements in an arbitrary PDS entries in X for CCCk must be divisible by pn . d Theorem 3.2.3 If C (d; k ) has any prime factor p > 2d, no PDSd exists for CCCk . k 2 P ROOF : Recall that the number of elements in any PDSd for CCCk must be Ck(d;k , an integer. For any ) n prime p > 2d, k can be written as = qp where n is a nonnegative integer and q a positive integer not 2k divisible by p. By the above, for any PDSd for CCCk , for any prime p > 2d, Cq(d;k must be an integer for ) some value of q not divisible by p. 3.2.4 Specific Cases Now, listing the calculated values of C (d; k ) in Table 5 (and noting that for k > 2d, C (d; k ) = C (d; 2d +1)), there are exactly two cases where C (d; k ) does not either contain a prime p > 2d or any prime p where k is not divisible by p — when d = 2 and k = 3 and when d = 3 and k = 5. This are small graphs, and an exhaustive search reveals that, in these cases as well, there does not exist a PDSd for CCCk . Hence, for all d > 1, no PDSd exists for CCCk . C OMMENT: When d = 5, C (d; 2d + 1) is 84. While a similar matrix argument exists, the one given here does not suffice. Also, it is not clear whether or the matrix argument could be generalized to handle arbitrary values of d. Thus, both arguments seem necessary. 3.3 Other Points of Interest: There may be some extended results which limit the number of ways in which a PDS1 may be constructed for C C Ck (k 6= 5). Essentially, the obvious rotations, reflections, and alternate choices for a and b (such 15 Table 5: Calculated values of C (d; k ) d=2 d=3 d=4 k=3 k=4 k=5 k=6 k=7 k=8 k=9 8 9 10 10 14 17 20 21 22 20 28 36 39 42 43 44 that k = 3  a + 4  b) using the above scheme might be shown to form all possible perfect dominating sets for d = 1 and certain values of k . No immediate application of this result is known to the authors. The existence and patterns generated by isolated points may have some application to the problem of finding generalized dominating numbers – the size of the minimal set of vertices which dominates the graph with distance d. Also, the matrix argument may be generalized to include other graphs, and has some application to the problem of dominating numbers. Work is currently in progress on Tori and related graphs. 3.4 Summary Standard perfect dominating sets exist and be easily constructed for cube-connected cycles of any order other than 5. However, for greater distances, no perfect dominating sets with more than one element exist for any cube-connected cycle. In fact, for d > 4, an isolated vertex exists, indicating that the size of a minimal dominating set will be much larger than optimal. (For 1 < d  4, a corresponding argument cannot be easily constructed.) Appendix A Complete Factorization of (p Since n) (p 1)( 1 YX n = ( n) 1 p 1 ( j) (ip )) j =0 i=0 it will be sufficient to show that each element in f 8 [ <X 1g :  p 1 (ipj ) : j 2 f0; 1; : : : ; n i=0 9 = 1g ; is irreducible in order to obtain a complete factorization. The following lemma will be useful for that purpose: 16 Lemma .0.1 For all integers m; n; i; j and prime p, ! m n  i X `=0 P ROOF : By induction on j . Noting that m n is an identity, assume that m n Now, by repeated application of m n ` `=0 n !  n ` ! p i m pj n `pj 1 ` `=0 p): (mod ! X ! ! i m i = `=0 ! X ! p p  ! m ipj n `pj ! X ! i i m ipj  for all m; n; i and prime p. Then, since m n ! i ` m n 0 ! pj `pj (mod m  ! ` ! 1 (mod 1 p) when i is not divisible by p, n ! pj m pj n pj + m pj n pj + p) ! ! (mod p): p); (mod the lemma follows. Lemma .0.2 For all nonnegative integers j , Pp 1 (ipj ) is irreducible when p is prime. i=0 P ROOF : This is a straightforward application of the Eisenstein irreducibility criteria. Let x =  1 and substitute. Immediately, the resulting polynomial is monic and has constant term p. All that is now required to show that it is! irreducible is show that all the other ! coefficients are ! divisible by! p. The nth coefficient is Pp ipj n 1 i=0 when n = `p for some `. When n j P equivalent to p 1 i=0 i ` ! or p `+1 = !  Pi` ipj n . Using the above identity, ipj n `p , j !  i ` =0 i ` n ! n) 1 is 8 [ <pX1 (ip ) f 1g :  : j 2 f0; 1; : : : ; n i=0 j `pj (mod modulo p. Thus, if n is not (p divisible by p. Thus, a complete factorization of (p 0 which is nonzero only p) and the nth coefficient is 1)pj , then the nth coefficient is 9 = 1g : ; References [1] Nathan Jacobson. Basic Algebra . W. H. Freemand and Company, New York, second edition, 1985. 17 [2] Nathan Jacobson. Basic Algebra I. W. H. Freemand and Company, New York, second edition, 1989. [3] Marilynn Livingston and Quentin F. Stout. Perfect Dominating Sets. Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, 1990). Congressus Numerantium, pages 187–203, 1990. 18