PERFECT DOMINATING SETS Ý Marilynn Livingston£ Quentin F

In Congressus Numerantium 79 (1990), pp. 187-203.

PERFECT DOMINATING SETS Ý Marilynn Livingston£ Quentin F. Stout Department of Computer Science Elec. Eng. and Computer Science Southern Illinois University University of Michigan Edwardsville, IL 62026-1653 Ann Arbor, MI 48109-2122

Abstract

G G A dominating set Ë of a graph is perfect if each vertex of is dominated by exactly one vertex

in Ë . We study the existence and construction of PDSs in families of graphs arising from the interconnection networks of parallel computers. These include trees, dags, series-parallel graphs, meshes, tori, hypercubes, cube-connected cycles, cube-connected paths, and de Bruijn graphs. For trees, dags, and series-parallel graphs we give linear time algorithms that determine if a PDS exists, and generate a PDS when one does. For 2- and 3-dimensional meshes, 2-dimensional tori, hypercubes, and cube-connected paths we completely characterize which graphs have a PDS, and the structure of all PDSs. For higher dimensional meshes and tori, cube-connected cycles, and de

Bruijn graphs, we show the existence of a PDS in inﬁnitely many cases, but our characterization d is not complete. Our results include distance d-domination for arbitrary .

1 Introduction

= ´Î; Eµ Î E i

Suppose G is a graph with vertex set and edge set . A vertex is said to dominate a

E i j i = j Ë Î

vertex j if contains an edge from to or if . A set of vertices is called a dominating

G Ë G

set of G if every vertex of is dominated by at least one member of . When each vertex of is

Ë G dominated by exactly one element of Ë , the set is called a perfect dominating set (PDS) of .

The size of a set of least cardinality among all dominating sets for G is called the domination G number of G and any dominating set of this cardinality is called a minimum dominating set for .It

is clear that a perfect dominating set for a graph is necessarily a minimum dominating set for it as

i d

well. These notions can be extended to d-domination, where a vertex is said to -dominate a vertex

i j G d d > ½

j if there is a path from to in of length at most .When , we will use the terminology

d Ë G

distance d PDS or perfect -dominating set to describe a subset of vertices of a graph such that

d Ë everyvertexof G is -dominated by a unique vertex in .

The concept of a perfect d-dominating set seems to have appeared ﬁrst in a paper by Biggs [5], d who introduced the term perfect d-code to denote what we call a perfect -dominating set. Biggs [5, 6]

was concerned with characterizing all perfect d-error correcting codes, and claimed that the proper

setting to study such issues is in the class of distance-transitive graphs. He deﬁned a distance-

Æ Ù; Ú ; Ü; Ý

transitive graph G as a connected graph with distance function such that whenever ,are

Æ ´Ù; Ú µ = Æ ´Ü; Ý µ G ´Ùµ = Ü

vertices of G for which , there is an automorphism of such that

´Ú µ = Ý and . Using algebraic techniques, Biggs derived an important necessary condition for

the existence of a perfect d-code in a distance transitive graph and applied his condition to several distance-transitive graphs including the classical hypercube graph.

£ Partially supported by National Science Foundation grant CCR-8808839 Ý Partially supported by National Science Foundation grant DCR-8507851 and an Incentives for Excellence Award from Digital Equipment Corporation

1 Later Bange, Barkauskas, and Slater [1], apparently unaware of Bigg’s work, defined the class of efficient dominating sets, which is exactly the same as the class of perfect 1-dominating sets. They concentrated on finding perfect dominating sets in trees, showing that there are linear-time algorithms that decide if a tree has a PDS, and if so then produce one [2]. Our motivation for studying the notion of perfect domination in graphs arose from our work involving resource allocation and placement in parallel computers [15]. To see how PDSs arise in this

context, suppose we have a parallel computer whose processors (pes) and interconnection network

=´Î; Eµ G are modelled by the graph G , where each pe is associated with a vertex of and a direct communication link between two pes is indicated by the existence of an edge between the associated vertices. Suppose, further, that we have a limited resource such as disks, I/O connections, or software modules, and we want to place a minimum number of these resource units at the pes, with at most

one per pe, and so that every pe is within a distance d of at least one resource unit. Finding such a

G G d placement involves constructing a minimum d-dominating set for the graph .If has a perfect - dominating set, this represents an optimal situation in which there is neither duplication nor overlap.

Even when a distance d PDS does not exist for a given graph, information about perfect dominating

sets for related graphs can be useful to help construct near optimum d-dominating sets. This is particularly true in graphs with quite regular structures, which are the graphs that arise from parallel computers.

Determining if an arbitrary graph has a dominating set of a given size is a well-known ÆÈ -

complete problem [8, 13]. Straightforward proofs can be used to show that it is also ÆÈ -complete to

decide if a graph has a PDS, and the problem remains ÆÈ -complete even if the graphs are restricted to 3-regular planar graphs. Thus the general problem of determining if a graph has a PDS is quite hard, but we show that for many signiﬁcant classes of graphs it is manageable. In Section 2 we consider whether perfect dominating sets exist for several classes of graphs which arise in the context of networks for parallel computers. These families include meshes, tori, trees, dags, series-parallel graphs, hypercubes, cube-connected cycles, cube-connected paths, and de Bruijn graphs. Except for hypercube graphs, none of these is distance transitive, and therefore the Biggs condition does not apply.

The existence of perfect d-dominating sets for trees, dags, and series-parallel graphs is studied in

½ d Section 2.1. For ﬁxed d , we give linear time algorithms that determine if a perfect -dominating set exists, and generate them when they do exist. In Section 2.2, we completely characterize all 2- and 3-dimensional meshes and 2-dimensional tori that possess a PDS and also characterize the structure of

the existing PDSs. For 2-dimensional meshes and tori we extend this to distance d PDSs for arbitrary

Distance d perfect dominating sets for hypercubes and hypercube related networks such as cube- connected cycles and cube-connected paths are considered in Section 2.3. For completeness we

include the complete characterization of distance d PDSs for hypercubes, which follows from the

results on perfect d-error correcting codes. Our characterization of the cube-connected cycles which

have a PDS and the structure of the existing PDSs is not complete, however. For, while we have k shown that there are inﬁnitely many dimensions k for which cube-connected cycles of dimension have a PDS, and that no PDS exists for dimensions 2 and 5, we do not know if there are inﬁnitely many dimensions for which a PDS does not exist. The situation for cube-connected paths is much simpler. In Section 2.3.3, we show no PDS can exist for cube-connected paths of even dimension, while for cube-connected paths of odd dimension, we construct the PDSs, which are unique up to isomorphism.

Graphs which are constructed from binary shift register sequences, called de Bruijn graphs, are B

considered in Section 2.4. For directed de Bruijn graphs, which we denote by k , we construct d

distance 1 PDSs for all k and show the existence of distance PDSs for inﬁnitely many values of

B k =½; ¾

k . Undirected de Bruijn graphs, denoted by ,haveaPDSfor , but we show that no PDS

=¿; 4; 5 d B k > 5 k ½ >d ½

exists for k . Whether distance PDSs exist for when and is not k

2 known.

2 Perfect Dominating Sets for Graphs

Æ ´G; Ú µ G =´Î; Eµ d Ú

Let d denote the set of vertices in the graph within a distance of vertex and

Ò ´G; Ú µ=jÆ ´G; Ú µj Ë d G fÆ ´G; Ú µ:Ú ¾ Î g

d d let d .If is a perfect -dominating set for then forms a

partition of Î and

Ò ´G; Ú µ=jÎ j:

d (1)

Ú ¾Ë

When G is regular or nearly regular, Equation ( 1) can be simpliﬁed, which provides a useful tool in d

combinatorial arguments for the existence of a PDS of G. We note also that if is at least the size of

G d the radius of G,then has a distance PDS. As we investigate the existence of perfect dominating sets in the families of graphs mentioned, we use many different techniques, depending on the particular graphs under consideration. For example, we introduce linear time algorithms to determine perfect dominating sets in trees, dags, and series- parallel graphs. For tori, hypercubes, cube-connected paths, and directed de Bruijn graphs, a mixture of algebraic and combinatorial methods are used. Several ad hoc methods are required for meshes, cube-connected cycles, and undirected de Bruijn graphs.

2.1 Trees

Ì Ù Ù Ì ¾

Let be a tree with two vertices ½ and that have a common parent, and suppose has the

Ì Ù Ù Ì ¾ property that any minimum dominating set for must contain ½ and .Then cannot have a

perfect dominating set. This means, for example, that a complete Ñ-ary tree of height greater than ½ 1 fails to have a PDS, for any Ñ> . On the other hand, as we shall see in the following theorem, there are no “forbidden” subgraphs that prevent perfect dominating sets in arbitrary trees or arbitrary

graphs.

d G G

Theorem 2.1 Given any graph G and any positive integer , there is a graph containing as an

d Ì d

induced subgraph, such that G has a distance PDS. Given any tree and any positive integer ,

Ì d

there exists a tree Ì containing as a subtree and which has a distance PDS.

=´Î; Eµ Ù G G Î [fÙg

Proof: Given a graph G ,let be a new vertex not in ,andlet have vertices ,

[ffÙ; Ú g : Ú ¾ Î g fÙg d G d

and edges E .Then is a distance PDS for for any .

= ½ d > ½

For the tree result we will give the proof for d as the proof for is similar. Let the Ö

tree Ì be given, and suppose is its root. We will proceed recursively, simultaneously building the

¼ ¼

Ë Ì = Ì Ë = Ö

tree Ì and a perfect dominating set, , as we go. Initially, and .If is a leaf, add a

Õ Ë Ö Ô Ë

child Õ to it and place in , otherwise, choose a child of , say , add it to . Recursively apply this

Ô ¾

procedure to the subtrees rooted at Ö ’s other children and to the subtrees rooted at ’s grandchildren.

Ì Ì Note that in the above construction of Ì , the only vertices added to were added to leaves of .

The question of whether an arbitrary tree Ì has a perfect dominating set can be answered in time

that is linear in the number of nodes of Ì . To see how this can be done, consider the following

algorithm.

Ì Ð ´Ú µ Ú ¾ Î Ð ´Ú µ

Let Î denote the set of vertices of and let denote the label of a vertex ,where

C; D ; Æ g Ú is a subset of f determined by the rules described below. Conceptually, the label of vertex

holds the information of the possible assignment of Ú as an element in some PDS that, at least up to

¾ Ð ´Ú µ Ú

that stage of the construction, is possible. Thus, if C then is already dominated (covered) by

¾ Ð ´Ú µ Ú

one of its children in some PDS construction to that stage, if D then is not dominated by any

Æ ¾ Ð ´Ú µ Ú of its children and Ú could be a dominator. Finally, if then all of ’s children are dominated

Algorithm 2.1 (PDS Finder for Trees)

Ì Ð ´Ú µ Ú ¾ Î

Let Î denote the set of vertices of and let denote the label of a vertex .

Ì Ð ´Ú µ=fD; Æ g

1. If Ú is a leaf of , initialize . Ì

2. Initialize ÒÓde = root of . ÒÓde 3. Traverse Ì in postorder, computing the label of as soon as the labels of all its children have been computed. Computation stops if the computed label is empty, for no PDS exists for

Ì .

but none of them is a dominator (i.e., Ú needs to be covered but cannot be a dominator itself). More

Ú Ð ´Ú µ C ¾ Ð ´Ú µ Ú

speciﬁcally, if all of Ú ’s children have labels, we compute ’s label as follows: if has

Ú C D ¾ Ð ´Ú µ

a child whose label contains D while the labels of the remaining children of all contain ;

Ú Æ ¾ Ð ´Ú µ C Ú

provided that Æ is in the label of each child of ; if is in the label of each child of ;if

´Ú µ= none of these hold, Ð .

The algorithm proceeds by computing the labels of the vertices of Ì in postorder and stops either

when the label of a node is determined to be empty, in which case Ì has no PDS, or when the label

of the root is determined to be nonempty. All PDSs of Ì can be generated by using the algorithm, proceeding from the root in a depth-ﬁrst search traversal, selecting a compatible element from the label of each node. The computation of the labels is outlined in Algorithm 2.1.

We thus have shown the following result, which was also proven earlier in [2]. Ì

Theorem 2.2 Let Ì be a tree. Algorithm 2.1 determines whether has a perfect dominating set in Ì time proportional to the number of vertices of Ì . Moreover, all perfect dominating sets for are

found by this algorithm. ¾

´Ú µ A straightforward modiﬁcation of Algorithm 2.1 and of the deﬁnition of Ð provides an algo-

rithm to determine the existence of a distance d PDS.

d Ì d

Corollary 2.3 Let Ì be a tree. For ﬁxed , the question of whether has a perfect -dominating d set can be answered in time proportional to the number of vertices of Ì . Further, if a perfect -

dominating set exists then one can be determined in time proportional to the number of vertices of Ì . ¾

The above methods can also be used to give a linear time decision algorithm to determine whether

a directed acyclic graph (dag) has a distance d PDS [16]. Adapting work on series-parallel graphs [18, 23, 24], these methods extend to yield a PDS decision algorithm for them in time proportional to the size of the graphs [16].

2.2 Meshes and Tori

Å ´Ñ ;Ñ ;::: ;Ñ µ k Ñ ¢ Ñ ¢ ::: ¢ Ñ

½ ¾ ½ ¾

k k

Let k denote a -dimensional mesh of size ,wherewe

Ñ Ñ ::: Ñ k

½ ¾

will assume the labelling is such that k . The vertices of this -dimensional

k ´i ;i ;::: ;i µ ½ i Ñ ½ j k

½ ¾ j j mesh are identiﬁed as -tuples of integers k ,where for .An edge is present between two vertices if and only if their labels differ in only one component and that

difference is one.

k Ñ ¢ Ñ ¢ ::: ¢ Ñ Ì ´Ñ ;Ñ ;::: ;Ñ µ k

½ ¾ ½ ¾

k k

The -dimensional torus of size k , denoted by is the -

Å ´Ñ ;Ñ ;::: ;Ñ µ Ù =

½ ¾ k dimensional mesh k enhanced with wrap-around connections. Two vertices

´Ù ;Ù ;::: ;Ù µ Ú =´Ú ;Ú ;::: ;Ú µ

½ ¾ ½ ¾ k

k and of the torus have a wrap-around connection in component

Ø fÙ ;Ú g = f½;Ñ g j 6= Ø Ù = Ú

Ø Ø j j provided Ø and, for all , .

It is easy to characterize those 1-dimensional meshes and tori that have a distance d PDS. We state

these in the following.

Å ´Ñµ d d

Theorem 2.4 The 1-dimensional mesh ½ always has a distance PDS for any . The 1-

Ì ´Ñµ d Ñ ¼ÑÓd´¾d ·½µ ¾

dimensional torus ½ has a distance PDS if and only if .

Ì ´Ñ; Òµ

In the next theorem, we give a characterization of the 2-dimensional tori ¾ that have a d distance d PDS. Our proof of the characterization, which appears in [16], shows also that for each

there is only one distance d PDS, up to isomorphism.

Ì ´Ñ; Òµ d fÑ; Òg

Theorem 2.5 The 2-dimensional torus ¾ has a perfect -dominating set if and only if

is a member of

ff¾; 4dÔg; f4; ´4d ¾µÔg; f6 ; ´4d 4µÔg;: :: ;f¾d; ´¾d ·¾µÔg : Ô ½g [

¾ ¾

ff´¾d ·¾d ·½µÔ; ´¾d ·¾d ·½µÕ g : Ô; Õ ½g: ¾

Thus we see that the only 2-dimensional tori for which distance 1 perfect dominating sets exist

Ì ´¾; 4Ôµ Ì ´4; ¾Ôµ Ì ´5Ô; 5Õ µ Ô Õ

¾ ¾ are ¾ , ,and ,where and are positive integers.

While we have not completed the PDS characterization for all 3-dimensional tori, we have found

Ì ´¾; ¿Ô; 6Õ µ

several instances for which a PDS exists. For example, ¿ has a PDS for all positive integers

Ô Õ Ô ;Ô ;::: ;Ô Ì ´´¾k ·½µÔ ; ´¾k ·

½ ¾ ½ k

and . In addition, for arbitrary positive integers k , the torus

½µÔ ;::: ;´¾k ·½µÔ µ ¾

k has a PDS. Perfect dominating sets for these tori appear in [16].

d Ì ´Ñ; Òµ Ñ; Ò ¾d ·½

Any distance PDS for the 2-dimensional torus ¾ , with , can be used to

d Å ´½; ½µ

construct a distance PDS for the inﬁnite 2-dimensional mesh ¾ , where we think of the torus as being unrolled and copies of it placed, non-overlapping, to cover the mesh. In [3], Bange et al. noticed a periodic distance 1 PDS for the inﬁnite 2-dimensional mesh and used it to construct

dominating sets for 2-dimensional meshes of ﬁnite size. The periodic distance 1 PDS observed by

Ì ´5; 5µ

Bange et al. is also the PDS constructed in [16] for ¾ . In the following two theorems we characterize all 2- and 3-dimensional meshes of ﬁnite size that

possess a PDS.

Å ´Ñ; Òµ Ñ = Ò = 4

Theorem 2.6 The 2-dimensional mesh ¾ has a PDS if and only if either ,

Ñ; Òg ff¾; ¾Ô ·½g : Ô a Ô Ó×iØiÚeiÒØegeÖg

or f is a member of the set . Furthermore, the PDS for

Å ´4; 4µ f´½;¾µ; ´¾;4µ; ´¿;½µ; ´4; ¿µg Å ´¾; ¾Ô ·½µ Ë

¾ Ô ¾ , unique up to automorphism, is . The PDS for , ,

is also unique up to automorphism, where

Ë = f´½ · ´i ÑÓ d ¾µ; ½·¾iµ:¼ i Ôg: Ô

Proof: First, it is straightforward to show that the given sets are indeed perfect dominating sets for

the speciﬁed meshes.

Ë Å ´¾;Òµ ´½; ½µ Ë ´½; ¾µ ´¾; ½µ

Now, suppose is a PDS for ¾ .If is not in , then exactly one of and

´½; ¾µ Ë ´¾; ½µ ´¾; ½µ

must be in Ë .If is in then cannot be dominated without overlapping, and if is

´½; ½µ

in Ë then the result will be a dominating set automorphic to one including . We conclude that,

; ½µ ¾ Ë Ë

without loss of generality, ´½ . From this point, the remaining elements of are completely

½ ´ÑÓ d ¾µ Ë

determined and we ﬁnd that Ò and that is as described in the statement of the theorem.

Ì Å ´Ñ; Òµ Ñ; Ò > ¾ ´½; ½µ Ì

If denotes a PDS for ¾ ,where , then the assumption that is in leads

; ¾µ ´¾; ½µ Ì

to a contradiction. It follows that exactly one of ´½ and must be in , and without loss of

; ¾µ Ì ´¿; ½µ; ´¾; 4µ; ´4; ¿µ Ì Ò =4 generality we take ´½ to be in . This forces and to be in . Thus, if

Ì Å ´4; 4µ Ò>4 ´½; 6µ ¾ Ì ´¿; 5µ ¾

then isaPDSfor ¾ .If ,however,then , leaving undominated.

A characterization of perfect dominating sets for 3-dimensional meshes can be found by similar

methods, as the proof of the following theorem illustrates.

¾ Ñ Ñ Ñ Å ´Ñ ;Ñ ;Ñ µ

¾ ¿ ¿ ½ ¾ ¿

Theorem 2.7 Let ½ . The 3-dimensional mesh has a PDS if and

Ñ = Ñ = Ñ =¾:

¾ ¿

only if ½

Ë Å ´Ñ ;Ñ ;Ñ µ f´½;½;½µ; ´¾;¾;¾µg

½ ¾ ¿

Proof: Suppose is a PDS for ¿ . It is easy to check that is

Å ´¾; ¾; ¾µ

aPDSfor ¿ , and that it is unique up to automorphism. Thus, it remains to show that

Å ´Ñ ;Ñ ;Ñ µ Ñ ;Ñ ;Ñ

½ ¾ ¿ ½ ¾ ¿

¿ has no PDS for any other values of .

Ñ = Ñ =¾ Ñ > ¾ ´½; ½; ½µ ¾ Ë ´¾; ¾; ¾µ Ë

¾ ¿

Case 1: suppose ½ and .If ,then must be in which

; ½; ¿µ ´½; ¾; ¿µ ´½; ½; ½µ Ë

leaves ´¾ and undominated. Consider, then, the possibility that is not in .

; ½; ¾µ ´½; ¾; ½µ Ë ´½; ½; ½µ ¾ Ë We must have ´¾ and therefore in . This is equivalent to the case under

automorphism, however.

Ñ = ¾ Ñ > ¾ ´½; ½; ½µ ¾ Ë ¾

Case 2: suppose ½ and .If , then we consider which element of

´½; ¾; ¾µ ´½; ¾; ¿µ ¾ Ë ´¾; ½; ¾µ

Ë can dominate . The assumption that leads to the fact that is not

; ¾; ¾µ ¾ Ë ´¾; ½; 4µ Ë

dominated; the assumption that ´¾ leads to the conclusion that is in , leaving

; ½; ¿µ

´½ undominated.

Ñ ¿

Case 3: suppose ½ . A similar but more lengthy case-by-case analysis establishes that no

Å ´Ñ ;Ñ ;Ñ µ ¾

½ ¾ ¿

mesh ¿ has a PDS.

> ½

For distance d , we have completed the characterization of all 2-dimensional meshes that d have a distance d PDS and the characterization of all perfect -dominating sets for them. We include,

below, the 2-dimensional meshes that have a distance d PDS but the complete statements and proofs

of these results are given in [16].

Å ´Ò; Ñµ ¾ Ò Ñ d

Theorem 2.8 The 2-dimensional mesh ¾ , , has a distance PDS if and only if

one of the following holds:

· Ñ ¾d ·½

( i) Ò

= Ñ =¾d ·¾

( ii) Ò

d ·½ ´Ñ ÑÓ d ¾d ·¾ Òµ ¾f½;::: ;¾d ·¿ ¾Òg

( iii) Ò and ¾

We have shown that weaker results of a similar form hold for k -dimensional meshes, in that

Å ´Ñ ;::: ;Ñ ;Òµ d Ò d Ñ ·

½ ½

k ½

k has a perfect -dominating set for inﬁnitely many whenever

¡¡¡ · Ñ ´k ½µ Ò d

k , and there are straightforward formulas giving such and the perfect -

Å ´Ñ ;::: ;Ñ µ d

½ k

dominating sets [16]. However, it may be that k does not have a distance PDS

Ñ ;::: ;Ñ ¾d ·½ d>½ k>¾ ½ when k for and .

2.3 Hypercubes and Related Graphs

Here we investigate the existence of distance d perfect dominating sets in the hypercube graph in Sec- tion 2.3.1, and in two variations of the hypercube graph: the cube-connected cycles in Section 2.3.2, and the cube-connected paths in Section 2.3.3.

2.3.1 Hypercubes

k É ¾

The hypercube graph of dimension , which we denote by k , consists of vertices labeled as

k ½

k ¾ k binary k -tuples, and edges, where each edge joins two vertices whose corresponding -tuples differ in exactly one component. These graphs have a long and rich history [11] and have recently become an important architectural model for several commercial parallel computers such as the Intel

iPSC and the NCUBE machines. d

For a hypercube graph of dimension k , a distance perfect dominating set is precisely a perfect

k ¾ binary d-error-correcting code over an alphabet of symbols. As illustrated in the survey article of van Lint [25], these codes have been the object of great interest since their inception by Hamming [10] and Golay [9] in the late 1940’s. All perfect binary error-correcting codes are known, in fact, all per-

fect codes over alphabets which are ﬁnite ﬁelds are known [25]. Perfect binary single-error-correcting

k ·½ codes of length k exist if and only if is a power of 2, and when this is the case, can be constructed directly using algebraic methods such as those described in [4] or can be constructed recursively, as shown by Golay [9]. The only perfect, binary, multiple-error-correcting code is the 3-error correcting

Golay code of dimension 23. We state these results in terms of distance d PDS in the following.

k É d

Theorem 2.9 The -dimensional hypercube, k , has a distance perfect dominating set if and only

if one of the following conditions is satisﬁed.

( i) k

=½ k ·½

( ii) d and is a power of two

=¿ k =¾¿

( iii) d and . ¾

2.3.2 Cube-Connected Cycles

k CCC k

The cube-connected cycles network of dimension , which we denote by k , consists of a -

k k dimensional hypercube, each of whose ¾ “vertices” consists of a cycle of nodes [20]. For each

dimension, every cycle has a node connected to a corresponding node in the neighboring cycle in that CCC

dimension. In order to be more precise, suppose that we label the nodes of k as ordered pairs

i; «µ ½ i k « k « k

´ ,where and isabinary -tuple. At position in the -dimensional hypercube, the

´½;«µ; ´¾;«µ;::: ;´k; «µ

k -cycle of nodes are labelled in, say, clock-wise order around the cycle as .

i; «µ ´j; ¬ µ Ø i = j = Ø

Moreover, two nodes ´ and are joined by an edge in dimension if and only if

k « ¬ Ø CCC

and the -tuples and differ only in the -th component. Figure 1 illustrates ¿ .

k k ½

CCC k ¾ ¿k ¾ ¾k

From the description we can see that k has nodes, edges, a diameter of ,and

¿ each node has degree 3 for k . These are some of the properties which make cube-connected cycles an interesting alternative architecture for multicomputers.

Characterizing which cube-connected cycles have distance d perfect dominating sets appears to k

be difﬁcult. Our ﬁrst result shows that if a distance d PDS exists for a -dimensional cube connected k

cycle graph, then a distance d PDS will exist for all dimensions that are a multiple of .

CCC d CCC Ñ ½ Ñk

Theorem 2.10 If k has a perfect -dominating set, then so does for all and

¾d ·½

k .

CCC CCC : ´Û; ¬ µ 7! ´Û ÑÓ d k; «µ k

Proof: We form a projection of Ñk onto by ,where

« = ¬ ¨ ¬ ¨¡¡¡¨¬ ½ i k ¨

i i

i·k

·´Ñ ½µk

i for all ,andwhere denotes the exclusive or operation.

CCC CCC k

It is straightforward to verify that with this projection, Ñk is a covering space of such that

´i; «µ CCC ´Û; ¬ µ ´i; «µ b´k ½µ=¾c

given any point in k and any point mapping onto , the distance -

Û; ¬ µ b´k ½µ=¾c neighborhood of ´ is mapped isomorphically onto the distance -neighborhood of

´i; «µ d CCC Ë d CCC

k k

. To construct a distance PDS of Ñk ,let be a distance PDS of ,andlet

Ë = ´Ë µ Ë d CCC

k Ñk Ñk

Ñk . A straightforward argument establishes that is a distance PDS for ,

b´k ½µ=¾c ¾

since d . CCC

We summarize in the table below what we have learned about the existence of a PDS for k

k 9

for ½ .

=½ k =¾

The Table 1 entry for k clearly holds. The case is easy, too, for equation (1) becomes

¡jË j =¾¡ ¾

¿ , which is impossible.

k = ¿; 4 Ë Ë CCC

4 ¿

We verify the cases by exhibiting perfect dominating sets ¿ and for and

CCC CCC ¿

4 , respectively. A PDS for is

Ë = f´½; ¼¼¼µ; ´¾; ½½¼µ; ´¿ ; ½¼ ½µ ; ´½ ; ½½½µ; ´¾; ¼¼ ½µ; ´¿ ; ¼½ ¼µ g ¿

and is unique up to isomorphism. It was also shown in [21] that a perfect dominating set exists for

CCC CCC ½6 4 ¿ . Now, any perfect dominating set for contains elements, with one element of the

PDS in each 4-cycle at the hypercube nodes. The set

f´¾;«µ:« i× Óf Ó dd ÔaÖiØÝ g[f´4;«µ:« i× Óf eÚeÒ ÔaÖiØÝ g CCC

isaPDSfor 4 , as can be easily veriﬁed. CCC

A simple counting argument establishes the fact that 5 has no PDS. By equation (1), if a

PDS, say Ë , did exist, it must contain 40 elements. Thus, some hypercube node must have at least 2

elements of Ë in its 5-cycle, but that is impossible.

=6; 8; Table entries for k and 9 follow from Theorem 2.10.

3,101 3,111

2,101 2,111 1,101 1,111

3,001 3,011

2,001 2,011 1,001 1,011

3,100 3,110

2,100 2,110 1,100 1,110 3,000 3,010

2,000 2,010 1,000 1,010

Figure 1: Cube-Connected Cycles of Dimension 3.

k CCC

PDS for k 1 yes 2 no 3 yes 4 yes 5 no 6 yes 7 ? 8 yes

9 yes CCC

Table 1: Existence of a PDS for k

2.3.3 Cube-Connected Paths

k CCÈ

The cube-connected paths network of dimension , which we denote by k , is very similar to the cube-connected cycles except that the cycles at the hypercube nodes are replaced by paths of length

k . For each dimension, the path at a hypercube node is connected to the corresponding node in the

path located at the neighboring hypercube node in that dimension. As in the cube-connected cycles,

CCÈ ´i; «µ ½ i k « k

we label the nodes of k as ordered pairs ,where and isabinary -tuple. At

k k

position « in the -dimensional hypercube, the nodes of the path of length of nodes are labelled

;«µ; ´¾;«µ;::: ;´k; «µ ´i; «µ ´j; ¬ µ

´½ . Notice that two nodes and are joined by an edge in dimension

i = j = Ø k « ¬ Ø

Ø if and only if and the -tuples and differ only in the -th component. An example of CCÈ

¿ is pictured in Figure 2.

k k ½

CCÈ k ¾ ´¿k ¾µ¾

From the description of cube-connected paths, we see that k has nodes,

k ¾ ´½;«µ ´k; «µ edges, a diameter of ¿ , and node degrees 2 and 3, where nodes with labels and

have degree 2, and other nodes have degree 3. One advantage of the cube-connected paths over cube-

CCÈ CCÈ

·½ k

connected cycles is that a k has two disjoint copies of in it, while the corresponding CCC

statement for k is false. This permits some recursive algorithms to be developed more cleanly CCC

on the CCÈ than on the .

k CCÈ

Theorem 2.11 If is an odd positive integer, then k has a perfect dominating set which is

unique up to isomorphism and is given by

Ì = f´i; «µ:i ½ ´ÑÓ d 4µ aÒd « i× Óf eÚeÒ ÔaÖiØÝ g [

f´i; «µ:i ¿ ´ÑÓ d 4µ aÒd « i× Óf Ó dd ÔaÖiØÝg:

k CCÈ

If is even then k has no perfect dominating set.

k Ì CCÈ k Proof: In the case when is odd, the demonstration that k isaPDSfor is straightforward,

so we do not include it here.

Ë CCÈ ½ i k Ü i

Suppose is a perfect dominating set for k . For each ,let denote the number

´i; «µ « ´i; «µ

of elements of Ë with label ,where is arbitrary. Since any node of the form must be

í ½;«µ í ·½;«µ í; ¬ µ k

adjacent to a unique element of Ë of the form , ,or , where the binary -tuples

Øh

« ¬ i Ü

and differ only in their component, the i must satisfy the following system of equations:

¾Ü · Ü =¾

½ ¾

Ü · ¾Ü · Ü =¾

½ ¾ ¿

Ü · ¾Ü · Ü =¾

¾ ¿ 4

:::

Ü · ¾Ü · Ü =¾

k ¾ k ½ k

Ü · ¾Ü =¾

k ½ k

9 d

Let k denote the determinant of the matrix of coefﬁcients of the above system of equations.

d =¿ d =4 d =¾d d k ¿ d = k ·½ k ¾

¾ ¿

k ½ k ¾ k Since , and k for , it follows that for all . Thus, the above system has a unique solution. The uniqueness of the PDS, up to isomorphism, now

follows easily for k odd.

k = ¾Ñ ½ i Ñ

For k even, say , and, for , a solution to this system is obtained by taking

k k

Ü =´Ñ i · ½µ¾ =´¾Ñ ·½µ Ü = i¾ =´¾Ñ ·½µ

i ½ ¾i

¾ and . Since these are not integer values, we may

CCÈ k ¾

conclude that k has no PDS when is even.

2.4 Binary de Bruijn Graphs

B k

Let k denote the graph whose vertices are binary -tuples, and for which an edge is directed from

Ü Ü ::: Ü Ý Ý ::: Ý Ü Ü ::: Ü = Ý Ý ::: Ý

½ ¾ ½ ¾ ¾ ¿ ½ ¾

k k k ½ the vertex k to vertex provided that . This graph,

known as a (directed) binary de Bruijn graph or shift register graph, is illustrated in Figure 3 for =¿ k . Parallel computers based on de Bruijn graphs have recently been suggested as alternatives to

hypercubes [19].

B ¼; ½;::: ¾ ½

When it is notationally convenient, we will denote the vertices of k as ,usingthe j

decimal equivalent of their binary k -tuple. With this notation, vertex has an edge directed from it to

j ¾j ·½ ¾

vertex ¾ and to vertex , where these numbers are taken modulo .

B k ½

The following theorem shows that the graphs k have perfect dominating sets for all ,and

d k

that for any d, distance perfect dominating sets exist for inﬁnitely many . The theorem leaves k open the possibility there are distance d PDS for other values of , a possibility partially discussed in

Proposition 2.13.

½ k ´d ·½µÑ ´d ·½µÑ ½

Theorem 2.12 For any d and for a positive integer of the form or or

3,101 3,111

2,101 2,111 3,001 3,011 1,101 1,111 2,001 2,011

1,001 1,011

3,100 3,110

2,100 2,110 3,000 3,010 1,100 1,110 2,000 2,010

1,000 1,010

Figure 2: Cube-Connected Paths of Dimension 3.

10 001 011

000 010 101 111

100 110 =¿

Figure 3: de Bruijn Graph for k .

Ì = Ì = ::: = Ì = f¼g

½ ¾

( i) d ,

´d·½µÑ ½ ´d·½µÑ

Ì = Ì [fj :¾ j ¾ ½g

d·½µ´Ñ·½µ ½ ´d·½µÑ ½

( ii) ´ ,

´d·½µÑ

Ì = Ì [f¾ ½ × : × ¾ Ì g

d·½µÑ ´d·½µÑ ½ ´d·½µÑ ½

( iii) ´ .

Ì B k

Then the set k is a perfect dominating set for .

f¼g d B k d k

Proof: It is easy to check that the set is a distance PDS for k when .For of the

´d ·½µÑ ½ Ì d B k

form , the fact that k is a perfect -dominating set for follows by induction and the

« k d

j · ¬ d j ¼ j < ¾

fact that all vertices of the form ¾ are within distance of vertex for all and

« d ¼ ¬ < ¾ k ´d ·½µÑ

½ , .For of the form , the result follows by virtue of the fact that

Ì Ì d B Ì

k ½ k ½ k ½

k is the union of , which is a distance PDS for , and the “reﬂection” of found by

¾ ½

taking the exclusive or of its elements with the k -tuple binary representation of . This reﬂection

k ½

j ¾ ¾ provides a unique dominator for every vertex j such that .

It may be the case that the only distance d dominating sets are of the form given above, and in

B k ½

particular that no perfect 2-dominating sets are possible for k when is a multiple of 3. However, B

we have been unable to prove or disprove that statement. To see that none can exist for 4 ,weoffer

Æ ´B ;Úµ Ú = ¼; ½5 Ú = 7; 8 4

the following argument. First, ¾ has: 4 elements if ; 5 elements if ;6

= 5; ½¼

elements if Ú ; and 7 elements otherwise. The criterion supplied by equation (1) serves to

4·5·7 5·5·6

reduce the possibilities to just three, corresponding to the partitions 4·6·6, ,and .

4·6 ·6 Æ ´B ; 5µ Æ ´B ; ½¼µ

4 ¾ 4 The case corresponding to the partition cannot occur because ¾ and

overlap. The remaining two cases are settled similarly. We have thus proved the following.

B ¾

Proposition 2.13 The graph 4 has no perfect 2-dominating set.

£ £

B B j B

Let us consider the undirected analog of k , which we denote by . Vertex in is adjacent

k k

k ½ k

j; ¾j ·½; bj=¾c; ¾ · bj=¾c ¾

to vertices ¾ , where these numbers are taken modulo . To illustrate,

k ½ k k ½ k k

; ½ ¾ ¾ ½ ¾ ½; ¾ ¾ ¾ ½

vertex 0 is adjacent to vertices ¼ and ; vertex is adjacent to and .

k ½ k>½ k ½ k>½

Since the radius of B is when , it follows that a distance PDS exists for all .

£ £

½g B B k

On the other hand, although f isaPDSfor and , we have found no other values of for

½ ¾

which a PDS exists. In the proof of the following proposition, we give a case-by-case analysis which

=¿; 4; 5

shows that no PDS exists for k .

k =½; ¾

Proposition 2.14 The undirected deBruijn graph B has a perfect dominating set for ,but

=¿; 4; 5 has no perfect dominating set for k .

£ £

½g B B

Proof: The fact that f isaPDSfor and was noted above. For the remaining cases,

½ ¾

£ £ k

Ò ´B ; ¼µ = Ò ´B ; ¾ ½µ = ¿ Ú ½

½ , and it is easy to see that these are the only two vertices for which

k k

£ £ ¿

Ò ´B ;Úµ=¿ Ò ´B ;Úµ=4 ¾·¾ · ::: · ½

½ . Further, there are only two vertices for which , namely

k k

k ¾ ¾ k ½ ¿ k ½ ¾ k ¾

½·¾ · ::: ·¾ k ¾· ¾ · ::: ·¾ ½·¾ · ::: ·¾ k

¾ and if is odd and and if is

£ £

Ò ´B ;Úµ=5 Ë B k

even. The remaining vertices have ½ . Now, suppose isaPDSfor for some .Let

k k

Ü Ë Ò ´B ;Úµ=i Ü

½ i

i be the number of elements of for which . As we have just observed, must be 0

i =¿; 4; 5 Ü ½ Ò ´B ;Úµ=4 ½

unless . Moreover, 4 because the two elements with have overlapping k

neighborhoods. It follows that

¿ ¡ Ü ·4 ¡ Ü ·5¡ Ü =¾ :

4 5

¿ (2)

k = ¿ Ü = Ü = ½ 5

Suppose . Equation (2) cannot hold unless ¿ . Without loss of generality we

¾ Ë B f¼; ½; 4g

may assume ¼ .Thismeansthatanyvertexof whose neighborhood overlaps cannot

Ë = f¼; ¿g = f¼; 7g

be in Ë . This leaves only the possibilities and , neither of which are a PDS.

k =4 Ü = Ü =¾ ¼ ½5 Ë 5

If then Equation (2) cannot hold unless ¿ . With both and in ,weﬁndthat

f6; 9; ½½g

the remaining 2 elements of Ë must be from the set . In order that element 3 be dominated,

¾ Ë

we must have 6 , but neither of the neighborhoods of 9 and 11 are disjoint from the neighborhood £

of 6. Thus, B has no PDS.

k = 5 Ü = Ü = ½ 4

Now suppose . Equation (2) cannot hold unless ¿ , which means that one of

¼; ¿½g f½¼; ¾½g Ë ¼ ¾ Ë ½¼ ¾ Ë

f and one of must be in . Without loss of generality we assume .If ,

½5 ¾ Ë

then either 15 or 30 must be in Ë .If , then the only possibilities for the remaining 4 elements

f4; 6; 8; 9; ½¾; ½¿; ½7 ; ¾¾; ¾5g

of Ë come from the set . A careful analysis of neighborhoods shows that

¿¼ ¾ Ë

it is impossible to complete the description of Ë in this case. In case , the remaining 4 elements

f4; 6; 9; ½¾; ½7; ½9; ¾¾ g Ë

of Ë must come from the set . Again, it is impossible to complete .Wemay

Ë Ë ½5 ¾ Ë

conclude that 10 is not in Ë and that 21 is in . Again, one of 15 or 30 is in . In the case ,

f4; 6; 9; ½¾; ½7; ½8; ¾5g ¿¼ ¾ Ë

the remaining 4 elements of Ë must be in ; in the case , the remaining 4

f4; 6; 9; ½¾; ½7; ½8; ½9 g Ë

elements of Ë must come from . In both of these cases, we ﬁnd that cannot

£ ¾

be completed. Therefore no PDS exists for B . 5

3 Variations on PDS i

Domination, in general, can be thought of as a binary relation of the form “Ü dominates ”, where

¾ X i ¾ Á X Á

Ü , ,and need not be the same as . Perfect dominating sets can be deﬁned for this general

¾ X i ¾ Á

notion of domination as follows. We call a dominating set Ë perfect if each is dominated

¾ Ë

by a unique Ü . With this deﬁnition, a perfect dominating set is not necessarily of minimum

= Á size unless X and the domination is symmetric. For example, in vertex-edge domination on

undirected graphs, consider a path È of 3 vertices. The two end vertices form a PDS, but the center vertex forms a PDS also. In general, a perfect dominating set may not be of minimum size, although it is always a minimal dominating set.

Let us consider some of these variations on domination in relation to the families of graphs con-

> ½

sidered in Section 2. While these variations have extension to distance d , we will restrict our

=½ Ú

discussion to d . In vertex-to-edge domination, denoted ve-domination, each vertex dominates

Ú; Õµ all the edges of the form ´ . There is a simple characterization for all undirected graphs that have

a perfect Úe-dominating set.

Úe G

Theorem 3.1 The undirected graph G has a perfect -dominating set if and only if is bipartite.

G G =´Í [ Í ;Eµ E Í

¾ ½

If is bipartite, say ½ with each edge in incident with one vertex in and one

Í Í Í Úe G ¾

½ ¾

in ¾ , then either of the sets or is a perfect -dominating set for .

Ù; Ú µ Ú In edge-to-vertex domination, denoted by ev-domination, an edge ´ dominates vertex . Undirected graphs which have perfect dominating sets in this sense are also easily characterized.

eÚ G Theorem 3.2 The undirected graph G has a perfect -dominating set if and only if has a complete

matching, in which case the set of edges in the complete matching form a PDS. ¾

É eÚ

In k , for example, we see that a perfect -dominating set is the set of edges with one vertex in

k ½µ ´k ½µ

one ´ -dimensional subcube and the other vertex in a vertex-disjoint -dimensional subcube.

eÚ CCC CCÈ k

Similar constructions yield perfect -dominating sets for k and .

Ù; Ú µ

For edge-to-edge domination, denoted by ee-domination, an edge ´ dominates all edges of the

Ú; Ôµ form ´ . Although we do not have a complete characterization of graphs with perfect dominating sets of this type, we will list a few of the more immediate results. The techniques of Section 2.1

can be used to yield a linear time decision algorithm that decides whether a tree has a perfect ee-

dominating set. As a consequence of the fact that an undirected graph with a 4-cycle can have no ee

perfect ee-dominating set, we ﬁnd that meshes, tori, and hypercubes have no perfect -dominating

CCC CCC 4

sets for dimension 2 or more. It follows similarly that ¿ and have no perfect dominating

CCÈ ee

set of this type. On the other hand, it is easy to show that ¿ does have a perfect -dominating

set. eÚ In [14], Laskar and Peters study the domination numbers of arbitrary graphs for Úe-, -, and

ee-domination.

4 Related Questions

Throughout the paper we noted several families of graphs for which the characterization of those

with perfect d-dominating sets is incomplete. It would be of particular interest, however, to complete the characterization of graphs with perfect dominating sets for higher dimensional meshes and tori, cube-connected-cycles, and the undirected de Bruijn graphs. There is a closely related question which arises from the placement of several different types of resources in a parallel computer. In order to perform this placement optimally on a given inter-

connection network, one is interested in the existence of several disjoint perfect d-dominating sets

Ì ´5; 5µ

for the graph of the network. Consider, for example, the torus ¾ with one PDS given by

´½; ½µ; ´¾; 4µ; ´¿; ¾µ; ´4 ; 5µ ; ´5; ¿µg

f . Using this set and its translates, it is straightforward to check that

Ì ´5; 5µ É ¾

has the maximum possible of 5 disjoint PDSs. For the hypercubes k which have a PDS,

k =¾ ½ É

namely those with , a perfect code for k , together with its cosets, yield a partition of the

É d

nodes of k , where each cell of the partition is a PDS. Existence of multiple perfect -dominating sets for other families of graphs is largely unexplored. There are a number of interesting and quite general questions that involve constructing perfect dominating sets for a graph from a given subgraph, or involve determining subgraphs which prevent a graph from possessing a PDS. We mention just two of these here and refer the reader to [17] for further discussion of these. Question 1. What are the forbidden subgraphs of perfect domination for trees (or some other large class of graphs) for various notions of domination? In addition to the subgraph requirement, more speciﬁcations may be added, such as for trees, one may wish to specify that certain leaves of the subgraphs are also leaves of the tree.

For the ÚÚ-domination considered in this paper, Theorem 2.1 showed that there are no forbidden subgraphs. However, if one requires that certain nodes of the subgraph are leaves, then there are

forbidden subgraphs. For example, the complete binary tree of height ¾, where all the leaves of the Ì binary tree must be leaves of the tree Ì , is a forbidden subgraph if is to have a PDS.

In the case of Úe-domination, for example, recall that a graph has a PDS of this type if and only

if it is bipartite. Thus, we ﬁnd that a graph has a perfect Úe-dominating set if and only if it does

not contain an odd cycle. The forbidden subgraphs for perfect Úe-domination for the class of all undirected graphs is the set of odd cycles. In the case of edge-vertex domination, however, there are

= ´Î; Eµ

no forbidden subgraphs for the class of all graphs. For, given any graph G , consider the

G ¢ É Ú ¾ Î Ú

product graph ½ , in which corresponding to every vertex there is a vertex (in the copy

¼ ¼

Ú Ú fÚ; Ú g eÚ

of G) with an edge between and . The set of new edges forms a perfect -dominating set.

=´Î; Eµ Ë Î E

Question 2.GivenagraphG , and a subset of (or of ), is it possible to add edges to

Ë G G

G to make a perfect dominating set for some supergraph of ? ¼

Admittedly, Question 2 is open-ended as stated. One can impose additional constraints on G such ¼

as requiring that it be connected, say, or that it be planar. If we require that the graph G be connected,

¼ G and G and are undirected, then we have the following observations. To avoid trivialities, we

assume that G has at least two vertices. ¼

If vertex-to-vertex domination is used, we ﬁnd that a connected G can be created if and only if

each of the following conditions holds.

¯ jË j½

G Ë

¯ No vertex of is dominated by more than one vertex of G

¯ The number of undominated vertices in is at least as large as the number of isolated vertices Ë

of G that are in . ¼

If vertex-to-edge domination is used, then a connected G can be created if and only if each of the

following conditions holds.

¯ jÎ j ½ jË j½

G Ë

¯ Every edge of is dominated by exactly one vertex of .

¼ Ë If edge-to-vertex domination is used, then a connected G can be created if and only if is a

perfect matching for G. ¼

If edge-to-edge domination is being used, then a connected G can be created if and only if each

of the following conditions holds.

¯ jË j½

G Ë

¯ Every edge of is dominated by exactly one edge of

G Ë ¯ has only two vertices, or at least one vertex is not an endpoint of an edge of .

Proofs of the above observations will appear in [17].

References

[1] D. Bange, A. Barkauskas, and P. Slater, “Disjoint dominating sets in trees”, Sandia Labs. tech. report SAND 78-1087J (1987).

[2] D. Bange, A. Barkauskas, and P. Slater, “Efﬁcient dominating sets in graphs”, Applications of Discrete Math., R.D. Ringeisen and F.S. Roberts, eds., SIAM, Philad. (1988) pp. 189-199.

[3] D. Bange, A. Barkauskas, L.H. Host, and P. Slater, “Efﬁcient near-domination of grid graphs”, Congr. Numer. 58 (1987) pp. 83-92.

[4] E.R. Berlekamp, Algebraic Coding Theory, Aegean Park Press, CA. Revised 1984 edition.

[5] N. Biggs, “Perfect codes in graphs”, J. Comb. Theory (B) 15 (1973) pp. 289-296.

14 [6] N. Biggs, “Perfect codes and distance transitive graphs”, Combinatorics, Proc. Third British Comb. Conf. (1973) London Math. Soc. Lecture Notes 13 (F.P. McDonough and V.C. Mavron, eds.) pp. 1-8. [7] E.J. Cockayne, E.O. Hare, S.T. Hedetniemi, and T.V. Wimer, “Bounds for the domination number of grid graphs”, Congr. Numer. 47 (1985) pp. 217-228.

[8] M.R. Garey and D.S. Johnson, Computers and intractability: a guide to the theory of ÆÈ - completeness (1979) W. H. Freeman & Co. New York. [9] M.J.E. Golay, “Notes on digital coding”, Proc. I.R.I. (IEEE) 37 (1949) p. 657. [10] R.W. Hamming, “Error detecting and error correcting codes”, Bell System Tech. J. 29 (1950) pp. 147-160. [11] F. Harary, J.P. Hayes, and H. Wu, “A survey of the theory of hypercube graphs”, Comput. Math. Appl 15 (1988) pp. 277-289. [12] O. Heden, “Perfect codes in antipodal distance transitive graphs”, Math. Institute of Univ. of Stockholm preprint (1974).

[13] D.S. Johnson, “The ÆÈ -completeness column: an ongoing guide”, J. Algorithms 6 (1985) pp. 434-451. [14] R. Laskar and K. Peters, “Vertex and edge domination parameters in graphs”, Congr. Numer. 48 (1985), pp. 291-305. [15] M.L. Livingston and Q.F. Stout, “Distributing resources in hypercube computers”, Proc. 3rd Conf. on Hypercube Concurrent Computers and Appl. (1988) pp. 222-231. [16] M.L. Livingston and Q.F. Stout, “On problems of distributing resources in parallel computers”, in preparation. [17] M.L. Livingston and Q.F. Stout, “Constructability and perfect dominating sets”, in preparation. [18] J. Pfaff, R. Laskar, and S.T. Hedetniemi, “Linear algorithms for independent domination and total domination in series-parallel graphs”, Congr. Numer. 45 (1984) pp. 71-82. [19] D.K. Pradhan and M.R. Samatham, “The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI”, IEEE Trans. Computers 38 (1989) pp. 567-581. [20] F.P. Preparata and J. Vuillemin, “The cube-connected cycles: a versatile network for parallel computation”, Commun. Assoc. Comput. Mach., 24 (1981) pp. 300-309. [21] A. Reddy, P. Banerjee and S. Abraham, “I/O embeddings in hypercubes”, Proc. Int. Conf.

on Parallel Processing (1988) pp. 331-338. Ç

[22] D.H. Smith and P. Hammond, “Perfect codes in the graphs k ”, J. Comb. Theory (1975) [23] K. Takamizawa, T. Nishezeki, and N. Sato, “Linear-time computability of combinatorial problems on series-parallel graphs”, Jour. Assoc. Comput. Mach., 29 (1982) pp. 623-641. [24] J. Valdes, R.E. Tarjan, and E.L. Lawler, “The recognition of series-parallel digraphs”, SIAM Jour. Comp. 11 (1982) pp. 298-313. [25] J.H. van Lint, “A survey of perfect codes”, Rocky Mountain Jour. of Math., 5 (1975) pp. 199-224.