In Congressus Numerantium 105 (1994), pp. 116–128.
Constant Time Computation of
Minimum Dominating Sets
Marilynn Livingston
Quentin F. Stout
Dept. of Computer Science
Southern Illinois University
Edwardsville, IL 62026-1653
Elec. Eng. and Comp. Sci.
University of Michigan
Ann Arbor, MI 48109-2122
Abstract
Let G be a graph and let P (n) denote an element from a one-parameter family of graphs, such as a path
of length n, a cycle of length n, or a complete binary tree of height n. We are concerned with determining
minimum dominating sets of graphs of the form G P (n). Using dynamic programming and properties of
finite state spaces, we show a constant time algorithm to produce a minimum dominating set of G P (n),
for fixed G and all n, for the one-parameter families mentioned. Previous researchers had used similar
techniques but obtained only linear-time algorithms. We also show how a closed form expression can be
obtained for the minimum domination number of G P (n). We discuss extensions of the algorithm to the
determination of all minimum dominating sets for G P (n), and to related problems of coverings, packings,
and codes. In addition, we discuss algorithm extensions to several different types of domination, including
perfect domination, and to other ways of composing graphs.
Key Words: codes, covering, domination, packing, matching, perfect domination, grid graph, product
graph, mesh, torus.
1 Introduction
Let G = (V; E ) denote an undirected graph. A subset of D vertices is called a dominating set of G if for
every v 2 V
D there is some u 2 D such that (u; v) 2 E (G). Sometimes a dominating set is referred
to as a vertex-vertex cover. The minimum cardinality of the dominating sets of G is called the domination
number of G and is denoted by (G).
The general problem of determining (G) for a given graph G, and of finding a dominating set D of
G of this minimum cardinality, has been an active area of research for many years [HL90]. When properly
stated, this problem has been shown to be NP-complete [GJ79], and remains so even when G is restricted to
certain simple classes of graphs. One example of this is the family of grid graphs, formed from products of
paths, where Pn denotes the path with n vertices. The m n complete grid graph, Pm Pn , has vertex set
V = f(i; j ) j 1 i m; 1 j ng and an edge between pairs of vertices (i; j ) and (u; v) if and only
if ji uj + jj v j = 1. A grid graph is any subgraph of a complete grid graph. T. Leighton proved that
determining the domination number of an arbitary grid graph is NP-complete [Jo85]. The complexity of the
domination problem for complete grid graphs is not known, however.
Partially supported by NSF/ARPA grant CCR-9004727
1
M. Jacobson and L. F. Kinch [JK84] found closed form expressions for (Pm Pn ) for m = 2; 3; 4.
E. Cockayne, E. Hare, S. T. Hedetniemi, and T. Wimer [CHHW] reported that they had inductive proofs for
m = 2; 3 and all n, and for m = 4 with n = 4k. Using an IBM 3081 to perform exhaustive search, they
found exact values for Pm Pn for m = 4 and 4 n 10, m = 5 and 5 n 8, m = 6, and n = 10; 11,
and used 20 CPU hours to determine (P7 P7 ). In addition, they constructed elementary arguments to
establish the inequality
(n + n 3)=5 (Pn Pn) (n + 4n c)=5
2
2
where c is 16, 17, or 20, depending on the remainder of n modulo 5. E. Hare, S. Hedetniemi, and W.
Hare [HHH] gave a (n) algorithm for computing (Pm Pn ) for fixed m. Their algorithm was based
on dynamic programming techniques with an associated state table. Using an IBM 3081, one of their implementations took one and a half minutes to construct the state table and then one additional minute to
compute (P7 P300 ). Using a less memory-intensive implementation, they computed (P8 P8 ) in 2.5
minutes and (P8 P19 ) in approximately 7 minutes. In [SP87], H.G. Singh and R.P. Pargas described a
parallel implementation to compute the domination number of Pm Pn . They obtained results for m 9
and n 10. Time requirements became prohibitive for m = 10, even for the 16-node FPS T series hypercube. More recently, T.Y. Chang and W.E. Clark [CC93] gave a lengthy proof of a closed form expression
for (Pm Pn ) for m = 5; 6, thus extending the results of E. Hare [H89] to all n 1 for these values of
m.
In this paper we show how to use the properties of finite state spaces, together with dynamic programming, to produce a (1) time algorithm for computing (Pm Pn ) for fixed m. In fact, we show how
to obtain closed form expressions for (G Pn ) for fixed graph G and all n. Moreover, we show how to
explicitly describe a minimum dominating set for G Pn in terms of a regular grammar over states derived
from G. Using only an IBM PC, we have been able to rapidly replicate the earlier results mentioned above,
and obtain closed form expressions for (Pm Pn ) for even larger m than was previously considered.
Further details of the algorithm implementation, and tables of (Pm Pn ), will appear in [LS94].
Many domination-related concepts defined for arbitrary graphs enjoy easy solutions or at least fast algorithms when restricted to the family of trees. S.M. Hedetniemi, S.T. Hedetniemi, and R. Laskar [HHL] give
an extensive coverage of domination and domination-related algorithms for trees, most of which are linear
time algorithms. The approach we describe in this paper can be easily modified to determine minimum
dominating sets for (G P (n)) when P (n) is a complete t-ary tree of height n, for fixed t and all n.
Our approach can be adapted to allow different types of domination as well, such as perfect [LS90],
efficient [BBHS, BBS] and total domination [HL90], and still retain the (1) time complexity. We will
illustrate with an example of this in Section 3.2.
A closely related concept to dominating sets is that of packing. Let k be a positive integer. A subset
K V is called a k-packing of the graph G = (V; E ) if the distance between every pair of vertices in K
is greater than k . The k -packing number of G is the cardinality of the largest k -packing of G. Note that
the 1-packing number of G is also known as the independence number of G, the largest size of a subset of
nonadjacent vertices of G. E. Hare and W. Hare [HH91] gave a linear time algorithm for determining the
2-packing number of the complete grid graph Pm Pn for fixed m. Recently, D.C. Fisher [F93] determined
the 2-packing number of Pm Pn for all m and n. He established a recursive inequality which enabled him
to deal with the cases for m 8 with elementary arguments in the same spirit as those in [CHHW]. Several
cases had to be treated separately, some of which were handled with a branch and bound algorithm, others
by ad hoc arguments. Using the techniques outlined in this paper, we have found a closed form expression
for the 2-packing number of Pm Pn for each m < 9 and all n, which allowed us to produce a considerably
shorter and simpler determination of the 2-packing numbers for all m and n. Further, our techniques easily
extend to k -packings of G Pn for arbitrary G and k .
2
Figure 1: A dominating set for P2 P8
s1 s2 s3 s4 s5 s6 s7
! !
!
!
Figure 2:
l
l
S , states for P Pn
2
2
2 An Illustrative Example
Before launching into the general description of the method, we illustrate it with a small example, showing how to compute minimal dominating sets for P2 Pn . Consider the graph P2 P8 and let S =
f(2; 1); (1; 3); (1; 5); (2; 5); (1; 8); (2; 8)g be one of its dominating sets. (S is a minimal dominating set, but
is not a dominating set of minimum size.) This is shown in Figure 1, where each vertex in S is labeled with
a , and all other vertices are labeled with an arrow pointing to an element of S that dominates them.
At vertex (1; 4) of Figure 1 there is a choice as to which dominating set element to point to. We force a
specific choice through the following interpretation of the labels, which will be critical for the constructions
in this paper. Vertex (j; k ) labeled
l
means that vertex (j; k ) is in S ,
means that vertex (j; k ) is not in S but at least one of the vertices (j 1; k ) is in S ,
means that none of the vertices (j; k ); (j
1; k) are in S , but (j; k 1) is in S .
! means that none of the vertices (j; k); (j 1; k); (j; k 1) are in S , but (j; k + 1) is in S .
2.1 States
Note that if we consider any column of the graph we have a copy of P2 with its vertices labeled by elements
of f; l; ; !g. Such a labeling of P2 which can arise from a dominating set in P2 Pk for some k will
be called a state. There are 42 possible labelings, but some reflection upon the interpretation of the labels
shows that a labeling is a state if and only if it satisfies the following conditions:
(S–i) if one entry is , then the other entry is or l.
(S–ii) if one entry is l, then the other entry is .
We will let S2 denote the set of all labelings of P2 which satisfy these conditions. It is easy to verify that S2
has 7 elements, given in Figure 2, where we show the states as column vectors of length 2. Note that all of
these states occur in Figure 1.
We will be constructing dominating sets for P2 Pk+1 from dominating sets (and sets that nearly
nominate) for P2 Pk . To help in this, we use the notion of state transitions to describe which states are
possible for column k + 1, given a particular state for column k . In general, it is possible to go from state si
in column ` to state sj in column ` + 1 if and only if the following conditions hold for all rows p:
3
s1
s2
s3
s4
s5
s6
s7
s1 s2 s3 s4 s5 s6 s7
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
1
1
1
1
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
Figure 3: The State Transition Table T2
final
state
initial
state
Figure 4: The State Transition Graph for P2
(T–i) if si (p) =!, then sj (p) = .
(T–ii) if si (p) = , then sj (p) 6=!.
(T–iii) if sj (p) =
, then si (p) = .
This information can be presented in the form of a state transition table T2 , given in Figure 3. The state
transition table consists of 7 rows and 7 columns in which a 1 appears in row i, column j if it is possible to
go directly from state si to state sj , and a 0 otherwise.
States which could be the first column of a dominating set will be called initial states, denoted by I2 ,
while those which could be the last column will be called final states, denoted by F2 . Note that a state is
initial if and only if it has no
entries, while a state is final if and only if it has no ! entries. We see from
Figure 2 that I2 = fs1 ; s5 ; s6 ; s7 g, and F2 = fs4 ; s5 ; s6 ; s7 g.
Let G2 denote the directed graph (with loops) whose vertices are the states in S2 and whose edges are
determined by the condition that (si ; sj ) is an edge if and only if T2 (si ; sj ) = 1, for all si ; sj 2 S2 . We call
G2 the state-transition graph for the pair S2 ; T2, and illustrate it in Figure 4. Adding the identification of
the initial and final states, one has a precursor of a finite state automaton, though no alphabet has yet been
specified.
4
2.2 State Sequences
Let S be a dominating set of Gn = P2 Pn . S induces a labeling of the vertices of Gn with elements of
f; l; ; !g. The pair (Gn; S ) can be associated with the sequence of states = h 1 ; 2 ; : : : ; ni, where
column j of Gn has the state j in S2 induced by S . We call the state sequence induced by the pair
(Gn; S ) and express this by writing hGn; S i = . For example, for the graph and dominating set given in
Figure 1, the induced state sequence is hs5 ; s2 ; s6 ; s3 ; s7 ; s4 ; s1 ; s7 i.
For any state sequence = h 1 ; : : : ; n i induced by a dominating set S , we see that we must have
T2( i ; i+1 ) = 1 for i = 1; 2; : : : ; n 1. Furthermore, the first state 1 must be in I2, and the final state
corresponds to a path in G2 of length n 1, starting in I2 and ending in F2 .
n must be in F2 . Thus
Let U denote a set of vertices of G2 , and, for j 1, let P(U; j ) denote the set of all paths in G2 of
length j 1 which begin at a vertex in I2 and end at a vertex in U . We shall be particularly interested in the
case when U is a single state or the set of final states. If S is a dominating set of Gn , we see that hGn ; S i
determines an element p of P(F2 ; n). Conversely, if p is an element of P(F2 ; n), then the sequence of n
vertices on the path p corresponds to a sequence of states ha1 ; a2 ; : : : ; an i which can be identified with a
unique dominating set A of P2 Pn , where
A = f(i; j ) j
the element in the i th row of state aj is g:
We express this relationship by writing hha1 ; a2 ; : : : ; ak ii = A or, equivalently, hhpii = A. Thus,
there is a natural 1-1 correspondence between the dominating sets of Gn and the paths of length
n 1 in the state-transition graph G2 which begin at an initial state and end at a final state.
Now, if we define the weight of the path p, w(p), as the total number of entries in all the columns of its
associated state sequence ha1 ; a2 ; : : : ; an i, then we see that w(p) is just the cardinality of the set A = hhpii.
Furthermore, the above 1-1 correspondence maps the minimum dominating sets of Gn onto the minimum
weight paths in P(F2 ; n).
2.3 The Cost Matrix
We organize our computational process in a 7 n matrix, Cn , which we call the cost matrix. The element
Cn(i; j ), in row i and column j , contains the quantities c(i; j ) and f (i; j ). The cost c(i; j ) is defined as
c(i; j ) =
(
minimum weight of the elements of P(fsi g; j ) if P(fsi g; j )
otherwise.
1
6= ;
The quantity f (i; j ) informs us of the most recent state from which we have made the transition to the
present state. More specifically, if j 2 and a = ha1 ; a2 ; : : : ; aj i is an element of P(fsi g; j ) of weight
c(i; j ), then f (i; j ) “points” to the second last vertex on the path a. If there is more than one such state
sequence of weight c(i; j ), we choose the smallest such k . So, for j 2,
f (i; j ) = k;
= sk :
To complete the definition of f (i; j ), we set f (i; 1) = 0 for 1 i 7.
Before we describe the recursive relations that exist for c(i; j ) and f (i; j ), we need some additional
notation. For s 2 S , let
P red(s) = fw j w 2 S and T (w; s) = 1g:
where
aj
1
2
2
5
2
1
s1
s2
s3
s4
s5
s6
s7
0;0
1;0
1;0
1;0
1;0
1;0
2;0
2
1;0
1;5
1;6
2;7
2;5
2;5
2;1
3
4
5
6
7
2;4
2;5
2;6
2;7
2;3
2;2
3;2
2;4
2;5
2;6
3;7
3;3
3;2
4;1
3;4
3;5
3;6
4;7
3;3
3;2
4;1
4;4
3;5
3;6
4;7
4;3
4;2
5;1
4;4
4;5
4;6
5;7
4;3
4;2
5;2
Figure 5: The Cost Matrix C7
Then, since each element of P(si ; j + 1) can be viewed as consisting of an element of P(sk ; j ), for some
sk 2 P red(si ), with the edge from sk to si appended to the path, we have the following recursive relation
for the c(i; j ):
c(i; j + 1)
= w((si) + minfc(k; j ) j sk 2 P red(si)g;
w(si ) if si 2 I
c(i; 1) =
1
otherwise
for
j 1;
(1)
2
The cost matrix C7 is shown in Figure 5, where the entry in row i and column j is exhibited as c(i; j ); f (i; j ).
For j 2, let the sequence k1 ; k2 ; : : : ; kj 1 be defined in terms of f (i; j ) as follows:
= f (i; j ); and
(2)
kt
= f (kt; j t); for 1 t j 2:
Let hfi;j i denote the associated state sequence hsk 1 ; : : : ; sk1 ; si i. It follows that
hfi;j i = hfk1;j isi for j 1:
To illustrate this notation, let us find a minimum dominating set for P P from C . Arbitrarily choosing
s from among the four states in F with minimum c(i; 6) values, we use the f (i; j ) entries to find the state
sequence hf ; i = hs ; s ; s ; s ; s ; s i. The minimum dominating set for P P corresponding to this
minimum weight path is hhf ; ii = f(1; 2); (2; 2); (1; 5); (2; 5)g.
k1
+1
j
+1
2
4
6
7
2
46
1
7
4
1
7
4
2
6
46
2.4 Periodic Behavior
An examination of Figure 5 reveals that
c(i; 7) = c(i; 5) + 1
for all i. We will refer to this behavior of the columns of Cn as periodic, for, once two columns, j1 and
j2 have the property that c(i; j1 ) = c(i; j2 ) + b for some constant b and all 1 i 7, it will be the case
that columns k and k + jj1 j2 j must differ by this constant b for all k max(j1 ; j2 ). This relationship,
together with Equation 1, implies that
c(i; j ) =
(
c(i; 5) + b(j
c(i; 6) + b(j
5)=2c
5)=2c
6
if j 1
otherwise
(mod 2)
(Gn) = d n e+3 for n 5. A check of the cost matrix for 1 n 4
(P Pn) = n +2 1 for n 1:
Now, let us turn to the construction of a minimum dominating set for P Pn . The periodic behavior
of the c(i; j ) guarantees that
(
6) if j 0 (mod 2)
f (i; j ) = ff ((i;i; 7)
otherwise
for j 7 and all i. Thus, if j is even and j 8, then Equation 2 becomes
k = f (i; 6)
k t = f (k t ; 7 )
for = 0; 1 and t 1. Similarly, for j odd and j 7, we obtain
k = f (i; 7)
k t = f (k t ; 6 + )
for = 0; 1 and t 1. Consequently, the associated state sequences hfi;j i satisfy recurrence relations for
each i. For example, taking i = 5,
hf ;j i = hf ;j is s s s
for j 10. Let denote the sequence s s s s , then
hf ;j i = hf ; j
ib j = c for j 10:
Further, we may express the sequences hf ;j i for 6 j 9 as
hf ; i = hf ; is s s = s
hf ; i = hf ; is s = s s s s
hf ; i = hf ; is
= sss
hf ; i = s
for j 7, and all i. It follows that
completes the proof of that
2
5
2
2
1
2 +
2 +
1
2 +
1
1
2 +
5
5
4
2 6 3 5
2 6 3 5
5
5 6+(
6 mod 4)
(
6) 4
5
2
59
26
6 3 5
5
58
66
3 5
5 6 3 5
57
36
2
5
6 3 5
56
5
Thus, from the periodic behaviour of the cost matrix, we see that, for all n, not only can we give the
value of (P2 Pn ), but we can also describe explicitly a minimum dominating set for P2 Pn , namely
hhf5;nii.
3 The Algorithm
To extend our methods from P2 Pn to G Pn for an arbitrary graph G, we have to slightly generalize the
definition of state. A state is a labeling of the vertices of G with elements of f; l; ; !g, satisfying the
restrictions
(S0 –i) if a vertex is labeled , then its neighbors are all labeled or l.
(S0 –ii) if a vertex is labeled l, then at least one of its neighbors is labeled .
The definitions of state transition, final state, and initial states are all exactly as they were in Section 2.
In Algorithm 3.1 we outline our algorithm for the cost matrix, from which we can compute domination
numbers and dominating sets of minimal size. As can be seen, both the time and space complexity of this
algorithm depend only on the graph G, although each will be exponential in the size of G.
The proof that the periodic behavior must occur, forcing the loop in steps 5-13 to terminate, appears
in [LS94].
7
Algorithm 3.1 (Domination Algorithm)
The logical variable periodic remains false until we discover periodic behavior between two columns
and K2 of the cost matrix.
K1
1 Determine S , the set of states, I , the initial states, F , the final states, and N , the number of states.
2 Determine the state transition table T .
3 Compute column 1 of the cost matrix C(; 1).
4
j := 1,
periodic:= false
5 repeat
6
j :=j + 1.
7
for i := 1 to N do
Compute c(i; j ) and f (i; j )
8
1 do
If c(; j ) c(; t) is a constant vector then
for t := 1 to j
9
10
11
periodic := true.
12
K1 := t, K2 := j .
13 until periodic.
3.1 Main Theorem
When Algorithm 3.1 terminates, post-processing of the cost matrix gives the following result.
Theorem 3.1 Let G be an arbitrary graph. Then there are integer constants m 1, n0
0 i m 1, such that, for all n n0,
0, a 1, and bi,
(G Pn ) = abn=mc + bi; where i = n mod m:
Additionally, for all 0 i m 1, there are states si;j and si;k of G, where 1 j n + i, and
1 k m, such that, for any n n , the set
Sn = hhsi; ; : : : ; si;n0 i ; (si; ; : : : ; si;m )b n n0 =mc ii;
where i = n mod m, is a dominating set of G Pn , with cardinality (G Pn ).
1
0
0
1
1
1
1
+
(
)
Further, these constants and states are determined by Algorithm 3.1 in a time which depends solely on
G. 2
We note that, while Theorem 3.1 emphasizes the asymptotic behavior, we also obtain the corresponding
results for n < n0 during the running of the algorithm, and hence determine (G Pn ) for all n.
8
Figure 6: Finite State Automaton for Perfect Domination
3.2 Perfect Domination
While we do not have space to show all of the variations that can be solved by this approach, we will
outline another example. One form of domination is perfect domination, where a subset S of vertices V
of a graph G is a perfect dominating set if it is a dominating set, and for every pair of vertices u, v in S ,
N (u) \ N (v) = ;, where N (v) denotes the neighborhood set of v, i.e., v and all vertices adjacent to v. Not
all graphs have perfect dominating sets: the smallest counterexample is a square of 4 vertices.
To determine those k for which G Pk has a perfect dominating set, we need to modify the definitions of states and of transitions. Still using a vertex labeling with the labels f; l; ; !g, with the same
interpretations, we impose the following restrictions on states:
(S00 –i) if a vertex is labeled , then all of its neighbors are labeled l.
(S00 –ii) if a vertex is labeled l, then exactly one neighbor is labeled .
For state transitions, we impose the following restrictions on going from state si to sj , for all vertices
G:
(T00 –i)
(T00 –ii)
p of
si(p) =! if and only if sj (p) = .
si (p) = if and only if sj (p) = .
Figure 6 shows the automaton for G = P2 . From this, it is easy to see that P2 Pn has a perfect dominating
set if and only if (n mod 2) = 1. We also see that states in which a pair of neighbors are both labeled , or
are both labeled !, can never contribute to a solution because they can have no predecessor or successor,
respectively. Thus we can add the following restrictions to states without changing our results:
(S00 –iii) if a vertex is labeled !, then none of its neighbors are labeled !
(S00 –iv) if a vertex is labeled
, then none of its neighbors are labeled
Using this automaton specially constructed for perfect domination, we obtain:
Theorem 3.2 Let G be an arbitrary graph. Then there are integer constants n0 1, m 1, and (possibly
empty) sets I f1; : : : ; n0 1g and J f0; : : : ; m 1g, such that G Pn has a perfect dominating set
if and only if
n 2 I if n < n0
(n mod m) 2 J otherwise.
Further, these constants and sets can be determined by an algorithm whose running time depends solely on
G.
9
Proof: G Pn has a perfect dominating set if and only if there is a path of length n 1, starting at an
initial state and ending at a final state, in the state transition graph described above. It is well known that the
lengths of paths of accepting sequences in a finite state automaton can be written in the form given in the
theorem, and that they can be determined from the state transition graph. 2
4 Conclusion
We have shown that, for any graph G, there is a closed-form formula for (G Pn ) as a function of n,
and that our algorithm finds this formula in time depending only on G. Further, dominating sets of minimal
size can be given as a regular grammar over states derived from G. We showed this by reducing the original
problem to one involving paths in a state space, and then solving this automata problem for all n by utilizing
dynamic programming and the periodic nature of the solution. While others had also noted that a state space
and dynamic programming could be used for this problem, apparently none had noticed that the periodic
properties of finite state spaces could be exploited to eliminate the time dependence on n.
By changing the definitions of the states (perhaps including labels of edges) and their transitions, this
approach can be extended to a great many other problems involving domination and domination-related
concepts such as packings, coverings, matchings, etc. For example, we can solve problems such as perfect domination, domination involving distances greater than 1, domination of nonconvex regions such as
knight’s moves ([HH87]), independent domination, edge-edge domination, etc. For packings, we can solve
problems such as k -packings, vertex-disjoint (or edge-disjoint) packings of subgraphs, etc. Covers can
include vertex covers, edge covers, covers by subgraphs, etc.
We can also change some of the information kept along with the dynamic programing, and use it to
answer various counting problems. For example, we can develop formulas for the number of dominating
sets, number of minimal dominating sets, number of dominating sets of minimal size, number of perfect
dominating sets, etc. We can count number of matchings, number of perfect matchings, and so on. Other
variations, still producing closed-form solutions, include replacing Pn by complete t-ary trees of height n
(for fixed t), or by cycles of n vertices. One can also vary the definition of product used, allowing us to
analyze, for example, grid graphs where each vertex is connected to its eight nearest neighbors, rather than
its four nearest neighbors.
Several of the extensions mentioned above will be explored in [LS94] and subsequent papers. Those
papers will include more details of material outlined here, including tables of (Pm Pn ).
References
[BBHS] D.W. Bange, A.E. Barkauskas, L.H. Host, and P.J. Slater, “Efficient near-domination of grid
graphs”, Congressus Numerantium 58 (1987) 83–92.
[BBS] D.W. Bange, A.E. Barkauskas, and P.J. Slater, “Efficient dominating sets in graphs”, Applications
of Discrete Mathematics, R.D. Ringeisen and F.S. Roberts, eds., SIAM (1988).
[CC93] T.Y. Chang and W.E. Clark, “The domination numbers of the 5 n and 6 n grid graphs”, J. Graph
Theory 17 (1993) 81–107.
[CHHW] E.J. Cockayne, E.O. Hare, S.T. Hedetniemi and T.V. Wimer, “Bounds for the domination number
of grid graphs”, Congressus Numerantium 47 (1985) 217–228.
[F93]
D.C. Fisher, “The 2-packing number of complete grid graphs”, Ars Combinatoria 36 (1993) 261–
270.
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[GJ79] M.R. Garey and D.S. Johnson, Computers and Intractability. A Guide to the Theory of NPCompleteness. W.H. Freeman, San Francisco (1979).
[H89] E.O. Hare, “Algorithms for grid and grid-like graphs”. Ph. D. thesis, Dept. Comp. Sc., Clemson
University, Clemson, SC (1989).
[HH91] E.O. Hare and W.R. Hare, “k -Packing of Pm Pn ”, Congressus Numerantium 84 (1991) 33–39.
[HH87] E.O. Hare and S.T. Hedetniemi, “A linear algorithm for computing the knight’s domination problem
of a k n chessboard”, Congressus Numerantium 59 (1987) 115–130.
[HHH] E.O. Hare, S.T. Hedetniemi and W.R. Hare, “Algorithms for computing the domination number of
k n complete grid graphs”, Congressus Numerantium 55 (1986) 81–92.
[HL90] S.T. Hedetniemi and R.C. Laskar, “Bibliography on domination in graphs and some basic definitions of domination parameters”, Discrete Math. 86 (1990) 257–277.
[HHL] S.M. Hedetniemi, S.T. Hedetniemi, and R. Laskar, “Domination in trees: models and algorithms”,
Graph Theory with Applications to Algorithms and Computer Science, Y. Alavi, G. Chartrand, L.
Lesniak, D. Lick, and C. Wall, eds., (1985) Wiley, 423–442.
[JK84] M.S. Jacobson and L.F. Kinch, “On the domination number of products of graphs”, Ars Combinatoria 18 (1984) 33–44.
[Jo85] D.S. Johnson, “The NP-Completeness column: an ongoing guide”, J. Algorithms 6 (1985) 434–451.
[KYK] T. Kikuno, N. Yoshida, and Y. Kakkuda, “A linear algorithm for the domination number of a seriesparallel graph”, Discrete Appl. Math., 37 (1983) 299–311.
[LS90] M. Livingston and Q.F. Stout, “Perfect dominating sets”, Congressus Numerantium 79 (1990) 187–
203.
[LS94] M. Livingston and Q.F. Stout, “Constant time computation of properties of product graph families”,
in preparation
[SP87] H.G. Singh and R.P. Pargas, “A parallel implementation for the domination number of a grid graph”,
Congressus Numerantium 59 (1987) 297–311.
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