This paper is available online at http://www.math.nthu.edu.tw/tjm/
ALGORITHMIC ASPECT OF k-TUPLE DOMINATION IN GRAPHS Chung-Shou Liao¤ and Gerard J. Chang¤
Abstract. In a graph G, a vertex is said to dominate itself and all of its
neighbors. For a fixed positive integerk, the k-tuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the k-tuple domination problem in graphs from an algorithmic point of view. In particular, we give a linear-time algorithm for the2-tuple domination problem in trees by employing a labeling method.
1. INTRODUCTION
The concept of domination in graph theory is a good model for many location problems in operations research. In a graphG, a vertex is said todominate itself
and all of its neighbors. Adominating set of G = (V; E) is a subset D of V such that every vertex inV is dominated by some vertex in D. Thedomination number
°(G) is the minimum size of a dominating set of G. Domination and its variations have been extensively studied in the literature; see [3, 7, 8].
Among the variations of domination, thek-tuple domination was introduced in [6]; also see [7, p. 189]. For a fixed positive integerk, a k-tuple dominating set of
G = (V; E) is a subset D of V such that every vertex in V is dominated by at least k vertices ofD. The k-tuple domination number °£ k(G) is the minimum cardinality of a k-tuple dominating set of G. The special case when k = 1 is the usual domination. The case whenk = 2 was calleddouble domination in [6], where exact
values of the double domination numbers for some special graphs are obtained. The same paper also gives various bounds of double andk-tuple domination numbers in Received January 4, 2001; revised June 28, 2001.
Communicated by F. K. Hwang.
2000Mathematics Subject Classification: 05C69.
Key words and phrases: Domination, k-tuple domination, algorithm, tree, leaf, neighbor.
¤Supported in part by the National Science Council under grant NSC89-2115-M009-037 and by the Lee and MTI Center for Networking Research at NCTU.
terms of other parameters. Nordhaus-Gaddum type inequality for double domination was given in [5].
The purpose of this paper is to study the k-tuple domination problem from an algorithmic point of view. In particular, we give a linear-time algorithm for the 2-tuple domination problem in trees.
We note that not every graph has a k-tuple dominating set. In fact, a graph G has a k-tuple dominating set if and only if ±(G) + 1 ¸ k, where ±(G) is the minimum degree of a vertex in G. As any nontrivial tree has at least two leaves, we only consider2-tuple domination for trees.
To establish our algorithm, we employ a labeling method similar to those for variations of domination in tree-type graphs; see [2, 4, 9, 10, 11, 12, 13]. Sup-pose G = (V; E) is a graph in which every vertex v is associated with a label M(v) = (t(v); k(v)), where t(v)2 fB; Rg and k(v) is a nonnegative integer. The interpretation of the label is that we want to find a “dominating set”D containing all vertices u with t(u) = R (called required vertices) such that each vertex v is
dominated by at leastk(v) vertices in D. More precisely, an M-dominating set of
G = (V; E) is a subset D of V satisfying the following conditions: (M1) If t(v) = R, then v2 D.
(M2)jNG[v]\ Dj ¸ k(v) for all vertices v 2 V , where NG[v] =fvg [ fu 2 V : uv 2 Eg is the closed neighborhood of the vertex v.
TheM-domination number °M(G) is the minimum cardinality of an M-dominating set inG. Notice that 2-tuple domination is M-domination with M(v) = (B; 2) for all vertices v in V . Also, G has an M-dominating set, i.e., °M(G) is finite, if and only if jNG[v]j ¸ k(v) for all vertices v in V . For instance, if G contains exactly one vertexx, then °M(G) = 0 when M(x) = (B; 0), °M(G) = 1 when M(x)2 f(B; 1); (R; 0); (R; 1)g, and °M(G) =1 otherwise.
2. 2-TUPLEDOMINATION IN TREES
To give an algorithm for the 2-tuple domination problem in trees, we in fact establish one for theM-domination problem in trees. We believe that the approach has potential for other classes of graphs. We first give the following theorem which is the base of the algorithm. Notice that it works for general graphs.
Theorem 2.1. Suppose G = (V; E) is a nontrivial graph in which every vertex vhas a label M(v) = (t(v); k(v)). Let x be a leaf adjacent to y.
(2) If k(x) = 2 or k(y) = jNG[y]j; then °M(G) = °M0(G0) + 1;where G0 is obtained from G by deleting x and M0 is obtained from M by relabeling y with t0(y) = R and k0(y) = maxfk(y) ¡ 1; 0g.
(3) If t(x) = R and k(x) < 2 and k(y) < jNG[y]j; then °M(G) = °M0(G0) + 1; where G0 is obtained from G by deleting x and M0 is obtained from M by relabeling y with k0(y) = maxfk(y) ¡ 1; 0g.
(4) If M(x) = (B; 1) and k(y) < jNG[y]j; then °M(G) = °M0(G0); where G0is obtained from G by deleting x and M0 is obtained from M by relabeling y with t0(y) = R.
(5) If M(x) = (B; 0) and k(y) < jNG[y]j; then °M(G) = °M(G¡ x). Proof. (1) This follows from the definition of M-domination.
(2) SupposeD0 is a minimum M0-dominating set of G0. Then y 2 D0, since t0(y) = R. Hence, D = D0[fxg is an M-dominating set of G, since jNG[x]\Dj ¸ 2¸ k(x). Thus, °M0(G0) + 1 =jD0j + 1 = jDj ¸ °M(G).
On the other hand, suppose D is a minimum M-dominating set of G. Then x; y 2 D, since k(x) = 2 or k(y) = jNG[y]j. Hence, D0 = Dnfxg is an M0 -dominating set of G0, since y 2 D0 and jN
G0[y]\ D0j = jNG[y] \ Dj ¡ 1 ¸ maxfk(y) ¡ 1; 0g = k0(y). So, °M(G) =jDj = jD0j + 1 ¸ °M0(G0) + 1.
These complete the proof of°M(G) = °M0(G0) + 1.
(3) SupposeD0 is an M0-dominating set of G0. Then D = D0[ fxg is an M-dominating set of G, since jNG[x]\ Dj ¸ 1 ¸ k(x). Thus, °M0(G0) + 1 = jD0j + 1 = jDj ¸ °M(G).
On the other hand, suppose D is a minimum M-dominating set of G. Then Then x 2 D, since t(x) = R. Hence, D0 = Dnfxg is an M0-dominating set of G0, since jNG0[y]\ D0j = jNG[y] \ Dj ¡ 1 ¸ maxfk(y) ¡ 1; 0g = k0(y). So, °M(G) = jDj = jD0j + 1 ¸ °M0(G0) + 1.
These complete the proof of°M(G) = °M0(G0) + 1.
(4) SupposeD0 is a minimum M0-dominating set of G0. Then y 2 D0, since t0(y) = R. Consequently, D0 is an M-dominating set of G as M(x) = (B; 1). Thus,°M0(G0) =jD0j ¸ °M(G).
On the other hand, suppose thatD is a minimum M-dominating set of G. If x =2 D, then y 2 D, since k(x) = 1. And so, D is an M0-dominating set of G0. Therefore, °M(G) = jDj ¸ °M0(G0). We may now assume that x 2 D. Let D0 = Dnfxg. If y 2 D0 and jNG0[y]\ D0j ¸ k(y), then D0 is an M0 -dominating set of G0 and so °M(G) =jDj > jD0j ¸ °M0(G0). So now y 62 D0 or jNG0[y]\ D0j < k(y) · jNG[y]j ¡ 1 = jNG0[y]j. For the case when y 62 D0, let z = y; for the case when y 2 D0, choose a vertex z 2 NG0[y]nD0. Then, in
any case,y 2 D0[ fzg and so D0[ fzg is an M0-dominating set of G0. Hence, °M(G) =jDj = jD0[ fzgj ¸ °M0(G0).
These complete the proof of°M(G) = °M0(G0).
(5) SupposeD0is a minimumM-dominating set of G¡ x. Then D0is also an M-dominating set of G, since t(x) = B and k(x) = 0. Therefore, °M(G¡ x) = jD0j ¸ °M(G).
On the other hand, suppose thatD is a minimum M-dominating set of G. If x =2 D, then D is also an M-dominating set of G¡ x. Thus, °M(G) =jDj ¸ °M(G¡ x). We may now assume thatx2 D. Let D0= Dnfxg. If jNG¡ x[y]\D0j ¸ k(y), then D0 is an M-dominating set of G¡ x and so °M(G) =jDj > jD0j ¸ °M(G¡ x). So nowjNG¡ x[y] \ D0j < k(y) · jNG[y]j ¡ 1 = jNG¡ x[y]j. Choose a vertex z 2 NG¡ x[y]nD0. Then D0[ fzg is an M-dominating set of G ¡ x. Hence, °M(G) =jDj = jD0[ fzgj ¸ °M(G¡ x).
These complete the proof of°M(G) = °M(G¡ x).
Based on the theorem above, we have the following linear-time algorithm for theM-domination problem in trees.
Algorithm. Find an M-dominating set of a tree.
Input. A tree T = (V; E) in which each vertex v is labeled by M(v) = (t(v); k(v)). Output. A minimum M-dominating set D of T .
Method.
D Ã ;; T0Ã T ;
while (T0has at least two vertices) do choose a leaf x adjacent to y in T0; if (k(x) > 2 or k(y) > jNT0[y]j) then
stop since there is no M-dominating set; elseif (k(x) = 2 or k(y) = jNT0[y]j) then
t(y) = R and k(y) = maxfk(y) ¡ 1; 0g and D Ã D [ fxg; elseif f¤ now k(x) < 2 and k(y) < jNT0[y]j ¤g (t(x) = R) then
k(y) = maxfk(y) ¡ 1; 0g and D Ã D [ fxg;
elseif f¤ now t(x) = B, k(x) < 2, k(y) < jNT0[y]j ¤g (k(x) = 1) then t(y) = R;
T0Ã T0¡ x f¤ delete x from T0 ¤g; end while;
suppose the only vertex ofT0is x;
if (k(x) > 1) then STOP as there is no M-dominating set; elseif (t(x) = R or k(x) = 1) then D Ã D [ fxg.
ACKNOWLEDGMENTS We thank the referee for many useful comments.
REFERENCES
1. S. Arumugam and S. Velammal, Edge domination in graphs,Taiwanese J. Math. 2 (1998), 173-179.
2. G. J. Chang, Labeling algorithms for domination problems in sun-free chordal graphs,
Discrete Appl. Math. 22 (1988/89), 21-34.
3. G. J. Chang, Algorithmic aspects of domination in graphs, in: Handbook of
Combi-natorial Optimization, D.-Z. Du and P. M. Pardalos, eds., Vol. 3, 1998, pp. 339-405.
4. E. J. Cockayne, S. E. Goodman and S. T. Hedetniemi, A linear algorithm for the domination number of a tree, Inform. Process. Lett. 4 (1975), 41-44.
5. F. Harary and T. W. Haynes, Nordhaus-Gaddum inequalities for domination in graphs,
Discrete Math. 155 (1996), 99-105.
6. F. Harary and T. W. Haynes, Double domination in graphs,Ars Combin. 55 (2000), 201-213.
7. T. W. Haynes, S. T. Hedetniemi and P. J. Slater,Domination in Graphs: the Theory, Marcel Dekker, New York, 1998.
8. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination in Graphs: Selected
Topics, Marcel Dekker, New York, 1998.
9. S. F. Hwang and G. J. Chang, The edge domination problem,Discuss. Math. – Graph
Theory 15 (1995), 51-57.
10. R. Laskar, J. Pfaff, S. M. Hedetniemi and S. T. Hedetniemi, On the algorithm com-plexity of total domination,SIAM J. Alg. Discrete Methods 5 (1984), 420-425. 11. S. L. Mitchell and S. T. Hedetniemi, Edge domination in trees, in: Proceedings
Eighth S. E. Conference on Combinatorics, Graph Theory and Computing, Utilitas
Math., Winnipeg, 1977, pp. 489-509.
12. P. J. Slater,R-Domination in graphs, J. Assoc. Comput. Mach. 23 (1976), 446-450.
13. M. Yannakakis and F. Gavril, Edge dominating sets in graphs, SIAM J. Appl. Math.
38 (1980), 364-372.
14. H. G. Yeh and G. J. Chang, Algorithmic aspects of majority domination,Taiwanese
J. Math. 1 (1997), 343-350.
15. H. G. Yeh and G. J. Chang, Weighted connected domination and Steiner trees in distance-hereditary graphs, Discrete Appl. Math. 87 (1998), 245-253.
16. H. G. Yeh and G. J. Chang, Weighted k-domination and weighted k-dominating clique in distance-hereditary graphs,Theoret. Comput. Sci. 263 (2001), 3-8.
Chung-Shou Liao and Gerard J. Chang Department of Applied Mathematics National Chiao Tung University Hsinchu 30050, Taiwan
