A note on obtaining k dominating sets from a k-dominating function on a tree

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1 A note on obtaining k dominating sets from a k-dominating function on a tree Robert R. Rubalcaba a,, Peter J. Slater a,b a Department of Mathematical Sciences, University of Alabama in Huntsville, AL 35899, USA b Department of Computer Science, University of Alabama in Huntsville, AL 35899, USA r.rubalcaba@gmail.com Abstract It has been shown that every double dominating set of a tree T can be partitioned into two dominating sets. We show that for every k-dominating function f : V (T) {0,,..., j} there are k dominating sets for which each v V (T) is in at most f(v) of them. Introduction For a vertex v in V (G) in the graph G = (V, E), the open neighborhood N(v) of v consists of the set of vertices adjacent to v, and the closed neighborhood of v is N[v] = N(v) {v}. Vertex v is said to dominate each vertex in N[v], including itself, and S V (G) is a dominating set if N[S] = s S N[s] = V (G), that is, if every vertex in V (G) is dominated by at least one vertex in S. A dominating set S of a graph G with the smallest cardinality is called a minimum dominating set and its size, the domination number, is denoted by γ(g). In the following, let k be a positive integer. Harary and Haynes [9] defined a set S of vertices to be a k-tuple dominating set if every vertex is dominated at least k times. When k is, this is called double domination (as in [8, 9]). The k-tuple domination number γ k is the minimum cardinality of a k-tuple dominating set. For k =, the double domination number γ (G) is also denoted as dd(g). For a graph G = (V, E) and a function f : V (G) {0,,..., j}, if S V (G) then f(s) = v S f(v). A function f is a dominating function if for each closed neighborhood N[v] we have f(n[v]), and f is a k- dominating function if f(n[v]) k for each v V (G). The weight of a function f is wgt(f) = f(v (G)). Note that for j = the function f is

2 a dominating function if and only if it is the characteristic function of a dominating set. All of our terminology is consistent with that in Haynes, Hedetniemi and Slater [0, ]. Domke, Hedetniemi, Laskar and Fricke [4] define a {k}-dominating function as a function f : {,,...k} V which satisfies f(n[v]) k for all v V. The minimum weight of such a function is denoted by γ {k}. Note that {k}-dominating functions dominate each vertex at least k times, while taking on integer values from 0 to k, as opposed to k- dominating functions, which take only the values of 0 or. In this paper, generalizing the result in [] for k =, we show that for every k-dominating function f : V (T) {0,,..., j} there are k dominating sets for which each v V (T) is in at most f(v) of them. Disjoint dominating sets Papers concerned with disjoint dominating sets in graphs include [,, 3, 5, 3], whereas [6, 7] are concerned with finding minimum cardinality dominating sets whose intersection is as small as possible. Recently, Hedetniemi, Hedetniemi, Laskar, Markus and Slater [] introduced the disjoint domination number γγ(g) = min{ S + S : S, S are disjoint dominating sets of G} (S is a dominating set, S is a dominating set, and S S = ). For the graph H in Figure, note that the double domination number of H is dd(h) = 3, while γγ(h) = 3 + k. u k H u u w w v v w k v k Figure : Graph H with dd(h) = 3 and γγ(h) = 3 + k

3 It is observed in [] that for a tree T we have dd(t) = γγ(t) (Theorem below). For completeness, we include a proof of this result which is then generalized in Theorem 3. Theorem ([]). If S is a double dominating set of a tree T, then S can be partitioned as S = S S where each S i is a dominating set. Proof. Let x be an endpoint of T with neighbor y, and consider T to be rooted at x. Because N[x] S, we have {x, y} S. We can construct S and S, starting with S = S =, as follows. Let x S and y S. For each v V (T) the height of v is its distance from the root x, hgt(v) = dist(x, v). Let H i = {v V (T) : v S and hgt(v) = i}. We have H 0 = {x} and H = {y}. We will decide the membership of each v H i in S and S before considering the membership of the vertices in H i+. So, assume we have done so for H 0, H,...H t, let v H t with t, and let the parent of v be w. If w S i for i = or, we let v be in the other set, v S 3 i. If neither the parent w nor its parent z is in S, then N[w] S implies that at least two children of w are in S. If v is the first child of w considered from H t, let v S, otherwise let v S. If the parent w / S and w s parent z S i, put v in S 3 i. We claim that each S i is a dominating set for T. To see this, let u V (T). If u S and its parent is also in S, then u and its parent are in different S i s so each S i dominates u. If u S and its parent is not in S, then N[u] S implies that u has at least one child c S. Again, u and c are in different S i s, so each S i dominates u. Now suppose u / S. Assume the parent p of u is in S, and suppose p S i. N[u] S implies some child c of u is in S. Note that we have chosen c S 3 i because c s parent u / S and u s parent p S i. Hence, each S i dominates u. Assume the parent p of u is also not in S. Then at least two children of u are in S, one of which was first chosen to be in S, and all of the others are in S. Again, each S i dominates u. Theorem ([]). For any tree T on n vertices, dd(t) = γγ(t). Proof. Suppose γγ(t) = k = S S, where S and S are disjoint dominating sets. Clearly for each u V (T) we have N[u] S N[u] S, so N[u] (S S ), and dd(t) k. If dd(t) = k = S where S is a double-dominating set of T, then Theorem implies that γγ(t) k. 3 (j, k)-domination As defined in Rubalcaba and Slater [4], if j k a function f : V (G) {0,,...j} is a (j, k)-dominating function if f(n[v]) k for every v 3

4 V (G). The (j, k)-domination number γ j,k (G) is the minimum weight of a (j, k)-dominating function. Note that if j < k then every (j, k)-dominating function is also a (t, k)-dominating function for all j t k. Note that when j = k, γ k,k = γ {k}, and when j =, γ,k = γ k. Thus, we have γ k (G) = γ,k (G) γ,k (G) γ k,k (G) = γ {k} (G). Generalizing Theorem for which j = and k =, we have the following. Theorem 3. Let T be a tree. If f : V (T) {0,,..., j} is a (j, k)- dominating function with j k, then there are k dominating functions f i : V (T) {0, } for i k with k i= f i(v) f(v) for every v V (T), or equivalently, there are k dominating sets S, S,... S k with {S i : v S i } f(v) for every v V (T). Proof. Let f be a (j, k)-dominating function of T with j k. Let x be a vertex of degree one of T and y its neighbor, root tree T at x, and for each v V (T), as before, let its height be its distance from x, hgt(v) = dist(x, v). We construct dominating sets S, S,...S k starting with each S i =, where each v will be in f(v) of the S i s, and where hgt(v) < hgt(w) implies that the sets containing v will be determined before deciding which S i s will contain w. Let t = f(x), and if t let x be in S, S,...S t. Because f(n[x]) k we have f(y) k t. Let y be in sets S t+, S t+,... S k, and if f(x)+f(y) k = j > 0 then let y also be in any j disjoint sets from among S, S,... S t. Assuming we have decided which S i s will contain each vertex w with 0 hgt(w) t, we assign membership in the S i s for the vertices v with height hgt(v) = t as follows. As was done for adjacent vertices x and y, assignment of memberships in the S i s will be done so that, if u and v are any adjacent vertices, the number of S i s containing u or v will be min(f(u)+f(v), k). For each vertex w with hgt(w) = t that has at least one child, let p be the parent of w and let v, v,...v c be the children of w. Because f(n[w]) k we have c j= f(v j) k f(w) f(p) = r. Assume r. We can put v in min(r, f(v )) of the S i s that do not contain w or p. Let r = k f(w) f(p) f(v ). If r we can put v into min(r, f(v )) of the S i s that do not yet contain a vertex in N[w]. We can continue until every S i does contain a vertex in N[w]. That is, if r, then for each S i that does not contain w or p we can choose a v j to put in S i in such a way that no v j is placed into more than f(v j ) of the S i s. At this point for i k the set S i contains a vertex in N[w]. Continuing, for each v j which has yet to be placed into f(v j ) sets, when possible assign v j to another S i with w / S i, and if f(v j ) + f(w) = k + r with r then, after assigning v j to the k f(w) sets S i with w / S i, also assign v j to any r of the sets that contain w. Note that the above method of assigning memberships in the S i s provides the following. For any vertex u with parent z, if f(u) + f(z) k 4

5 then each S i with i k contains at least one of u and z, and if f(u)+f(z) < k then each S i with S i {u, z} = contains at least one child of u. It follows that each S i is a dominating set and that for each vertex v V (T) we have {S i : v S i } f(v). Corollary 4. For any tree T, γ j,k (T) k γ(t). 4 Beyond trees The graphs in Figure (a) and Figure (b) serve as counterexamples to possible extensions of Theorem 3. The 4 grid graph has γ(g,4 ) = 3, however the (, 5)-dominating function f depicted in Figure (a) has weight < 5 γ(g,4 ). So clearly we cannot get five dominating sets to satisfy Theorem 3. The (, 3)-dominating function depicted in Figure (b) for C 4 also shows that Theorem 3 cannot be extended to bipartite graphs. Figure : (a) The 4 grid graph G,4 has a unique (, 5)-dominating function and (b) C 4 has a unique (, 3)-dominating function A graph is chordal if there are no induced cycles of length greater than three. A graph is strongly chordal if it is chordal and contains no induced trampolines (in Figure, if we let k = the graph H is a trampoline on 6 vertices, also called a 3-sun). Note that all trees are strongly chordal. Theorem 3 cannot be extended to strongly chordal graphs either, since the strongly chordal graph below has γ(p) = and a (, 5)-dominating function with weight 6 < 5 γ(p). While the (, 5)-dominating function of G,4 cannot be decomposed into five dominating sets, it can be decomposed into four dominating sets (see Figure 4). While the (, 3)-dominating function of C 4 cannot be decomposed into three dominating sets, it can be decomposed into two dominating sets. The strongly chordal graph P in Figure 3 has a (, 5)-dominating function which can be decomposed into three dominating sets. 5

6 P Figure 3: The (, 5)-dominating function cannot be decomposed into five dominating sets Figure 4: Obtaining four dominating sets from the (, 5)-dominating function 6

7 References [] D. W. Bange, A. E. Barkauskas, and P. J. Slater, Disjoint dominating sets in trees, Sandia Laboratories Report SAND J (978). [] D. W. Bange, A. E. Barkauskas, and P. J. Slater, A constructive characterization of trees with two disjoint minimum dominating sets, Congressus Numerantium (978) 0. [3] E. J. Cockayne and S.T. Hedetniemi, Disjoint independent dominating sets in graphs, Discrete Mathematics 5 (976) 3. [4] G. S. Domke, S.T. Hedetniemi, R. C. Laskar, and G. Fricke, Relationships between integer and fractional parameters of graphs, in Y. Alavi, G. Chartrand, O. R. Oellermann and A. J. Schwenk (Eds.), Graph Theory, Combinatorics, and Applications, Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, vol., pages , (Kalamazoo, MI 988), Wiley Publications, 99. [5] J. E. Dunbar, S. M. Hedetniemi, S. T. Hedetniemi, D. P. Jacobs, J. Knisley, R. C. Laskar, and D. F. Rall, Fall colorings of graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 33 (000) [6] D. L. Grinstead and P.J. Slater, On minimum dominating sets with minimum intersection, Discrete Mathematics 86 (990) [7] D. L. Grinstead and P.J. Slater, On the minimum intersection of minimum dominating sets in series-parallel graphs, in Y. Alavi, G. Chartrand, O. R. Oellermann and A. J. Schwenk (Eds.), Graph Theory, Combinatorics, and Applications, Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, vol., pages , (Kalamazoo, MI 988), Wiley Publications, 99. [8] F. Harary and T. W. Haynes, Nordhaus Gaddum inequalities for domination in graphs, Discrete Mathematics 55 (996) [9] F. Harary and T. W. Haynes, Double domination in graphs, Ars Combinatoria 55 (000) 0 3. [0] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 998. [] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater (Eds.), Domination in graphs: Advanced Topics, Marcel Dekker, Inc., New York,

8 [] S. M. Hedetniemi, S. T. Hedetniemi, R. C. Laskar, L. Markus and P. J. Slater, Disjoint dominating sets in graphs, submitted. [3] V. R. Kulli and S. C. Sigarkanti, Inverse domination in graphs, National Academy of Science Letters 4 (99) [4] R. R. Rubalcaba and P. J. Slater, Efficient (j, k)-domination, to appear in Discussiones Mathematicae Graph Theory. 8