AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 52 (2012), Pages 281–292 Minimal dominating sets in maximum domatic partitions S. Arumugam∗ K. Raja Chandrasekar National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH) Kalasalingam University Anand Nagar, Krishnankoil-626126 India [email protected] [email protected] Abstract The domatic number d(G) of a graph G =(V,E)isthemaximumor- der of a partition of V into dominating sets. Such a partition Π = {D1,D2,...,Dd} is called a minimal dominating d-partition if Π contains the maximum number of minimal dominating sets, where the maximum is taken over all d-partitions of G. The minimal dominating d-partition number Λ(G) is the number of minimal dominating sets in a minimal dominating d-partition of G. In this paper we initiate a study of this parameter. 1 Introduction By a graph G =(V,E), we mean a finite, undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n and m respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [4]. Let G =(V,E) be a graph. A subset S of V is called a dominating set of G if every vertex not in S is adjacent to at least one vertex in S. A dominating set S is called a minimal dominating set in G if no proper subset of S is a dominating set of G. The minimum cardinality of a minimal dominating set of G is called the domination number of G and is denoted by γ(G). An excellent treatment of fundamentals of domination is given in the book by Haynes et al. [7] and survey papers on several advanced topics are given in the book edited by Haynes et al. [8]. A domatic partition of G is a partition of V (G) into classes that are pairwise disjoint dominating sets. ∗ Also at School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia. 282 S. ARUMUGAM AND K. RAJA CHANDRASEKAR The domatic number of G is the maximum cardinality of a domatic partition of G and it is denoted by d(G). The domatic number was introduced by Cockayne and Hedetniemi [5]. For a survey of results on the domatic number and its variants we refer to Chapter 13 in [8] by Zelinka. In the context of vertex coloring Arumugam et al. ([1] and [3]) investigated the problem of determining the maximum number of maximal independent sets (equiv- alently, independent dominating sets) in a minimum coloring of G.Inthispaper we investigate an analogous problem, namely, the maximum number of minimal dominating sets in a maximum domatic partition of G. We need the following definitions and theorems. Theorem 1.1. [5] For any graph G of order n, 1. d(G) ≤ δ(G)+1,whereδ(G) denotes the minimum degree of G. 2. d(G) ≤ n/γ(G). Definition 1.2. AgraphG for which d(G)=δ(G) + 1 is called domatically full. Definition 1.3. Let S ⊆ V and u ∈ S. Then a vertex v is called a private neighbor of u with respect to S if N[v] ∩ S = {u}. If further v ∈ V − S,thenv is called an external private neighbor of u. Definition 1.4. Let A and B be two disjoint nonempty subsets of V .Wedenote by [A, B], the set of all edges in G with one end in A and other end in B.Agraph G =(V,E)issaidtohaveabijective matching if there exists a nonempty subset A of V such that [A, V − A] is a perfect matching in G. Clearly if G has a bijective matching, then n is even and |A| = n/2. Definition 1.5. Given an arbitrary graph G,thetrestled graph of index k, denoted Tk(G), is the graph obtained from G by adding k copies of K2 foreachedgeuv of G and joining u and v to the respective end vertices of each K2. Definition 1.6. The corona of two graphs G1 and G2 is defined to be the graph th G = G1 ◦ G2 formed from one copy of G1 and |V (G1)| copies of G2 where the i th vertex of G1 is adjacent to every vertex in the i copy of G2. Theorem 1.7. [10] For two arbitrary graphs G and H, d(G ◦ H)=d(H)+1. Definition 1.8. The join graph G1 + G2 is obtained from two graphs G1 and G2 by joining every vertex of G1 with every vertex of G2. Definition 1.9. The cartesian product of G and H, written GH, is the graph with vertex set V (GH)={(u, v):u ∈ V (G) and v ∈ V (H)} and edge set E(GH)={(u, v)(u,v):u = u and vv ∈ E(H) or v = v and uu ∈ E(G)}. Definition 1.10. The n-cube Qn is the graph whose vertex set is the set of all n-dimensional boolean vectors, two vertices being joined if and only if they differ in exactly one coordinate. We observe that Q1 = K2 and Qn = Qn−1K2 if n ≥ 2. MINIMAL DOMINATING SETS 283 Theorem 1.11. [8] For any positive integer k, the hypercube Q2k−1 is a regular domatically full graph. Definition 1.12. A decomposition of a graph G is a family C of subgraphs of G such that each edge of G is in exactly one member in C. Observation 1.13. Any graph G admits a decomposition C of G such that each member of C is either a cycle or a path. Theorem 1.14. [8] For the Petersen graph P , d(P )=2. 2MainResults Let G be a graph with domatic number d and let Π = {D1,D2,...,Dd} be a domatic partition of G.IfD1 is not a minimal dominating set of G, then there exists v ∈ D1 such that D1 −{v} is also a dominating set of G.WenowreplaceD1 by D1 −{v} and Dd by Dd ∪{v}. By repeating this process we obtain a domatic partition of G such that D1 is a minimal dominating set of G. In fact we can apply the same process of transferring elements from D2,D3,...,Dd−1 to Dd and obtain a domatic partition ∗ Π = {S1,S2,...,Sd} of G such that S1,S2,...,Sd−1 are minimal dominating sets of G. This observation motivates the following definition. Definition 2.1. Let G be a graph with domatic number d(G). A domatic partition Π={D1,D2,...,Dd} is called a d-partition of G. A d-partition Π is called a minimal dominating d-partition if Π contains maximum number of minimal dominating sets, where the maximum is taken over all d-partitions of G. The minimal dominating d-partition number Λ(G) is the number of minimal dominating sets in a minimal dominating d-partition of G. It follows immediately from the definition that Λ(G)=d(G) − 1ord(G). Definition 2.2. AgraphG is said to be class 1 or class 2 according as Λ = d − 1 or Λ = d. A dominating set in a graph G can be thought of as a “secure set” in the sense that if a guard is placed at each vertex of a dominating set D of a graph G,thenevery vertex of G comes under the surveillance of at least one guard. If the dominating set D is minimal, then every guard in the troop is essential for the security of the graph. In this process of security, placing the guards always at the some set of vertices is not desirable. Hence if we can identify a collection C of disjoint minimal dominating sets in G, then at any point of time we can place a troop of guards, one at each vertex of a chosen dominating set from C. In particular if we can find a domatic partition {D1,D2,...,Dd} of G such that every Di is a minimal dominating set of G, then we obtain d different options for placing the guards in G. A graph that admits such a partition is of class 2. Otherwise we get a collection of d − 1 disjoint minimal dominating sets, each of which gives a possible options for the placement of a troop of guards. 284 S. ARUMUGAM AND K. RAJA CHANDRASEKAR Examples 2.3. 1. Any graph with domatic number n is isomorphic to Kn and is of class 2. 2. Any graph with domatic number n−1 is isomorphic to Kn −e and is of class 2. 3. For the corona graph H = G ◦ K1, d(H)=2. Also Π = {V (G),V(H) − V (G)} is a d-partition of H and every member of Π is a minimal dominating set of H. Hence Λ(H)=2andH is of class 2. 4. The complete bipartite graph Km,n with m ≤ n is of class 2 if and only if m ∈{1, 2} or m = n. Theorem 2.4. Let G be a graph of order n ≥ 3 with d(G)=n − 2. Then G is of class 2 if and only if G is isomorphic to K3 + Kn−3 or H + Kn−4, where H ∈{P4,C4, 2K2}. Proof. Suppose G is of class 2. Let {V1,V2,...,Vn−2} be a domatic partition of G such that each Vi is a minimal dominating set of G. Suppose |V1| =3.Then|Vi| =1, for all i =2, 3,...,n− 2. Hence G = V1 + Kn−3. Since V1 is a minimal dominating set, it follows that V1 is independent and hence G is isomorphic to K3 + Kn−3.