Gallai-type theorems and domination parameters

Let 7(G) denote the minimum cardinality of a dominating set of a graph G = (V,E). A longstanding upper bound for 7(G) is attributed to Berge: For any graph G with n vertices and maximum degree A(G), 7(G) <~ n A(G). We eharacterise connected bipartite graphs which achieve this upper bound. For an arbitrary graph G we furnish two conditions which are necessary if 7(G) + A(G) = n and are sufficient to achieve n 1 ~< 7(G) + A(G) <~ n. We further investigate graphs which satisfy similar equations for the independent domination number, i(G), and the irredundance number ir(G). After showing that i(G) <~ n A(G) for all graphs, we characterise bipartite graphs which achieve equality. Lastly, we show for the upper irredundance number, IR(G): For a graph G with n vertices and minimum degree 6(G), IR(G) <<, n 6(G). Characterisations are given for classes of graphs which achieve this upper bound for the upper irredundance, upper domination and independence numbers of a graph. 1. I n t r o d u c t i o n Let G = (V ,E) be a graph. For any vertex x E V we define the neighbourhood o f x, denoted N(x) , as the set of all vertices adjacent to x. The closed neighbourhood o f x, denoted N[x], is the set N ( x ) U {x}. For a set of vertices S, we define N ( S ) as the union of N(x ) for all x E S, and N[S] = N ( S ) U S. If x E S, a private neighbour o f x with respect to S is a vertex v EN[S] N[S {x}]. The degree of a vertex is the size of its neighbourhood. The maximum degree of a graph G is denoted A(G) and the minimum degree is denoted by 6(G). In this paper, n will denote the number of vertices in a graph. * Corressponding author. E-mail: [email protected]. 0012-365X/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PH SO0 12-365X(97)0023 1-2 238 G.S. Domke/D&crete Mathematics 1671168 (1997) 237-248 A set S _C Y is said to be independent if every pair of vertices in S is nonadjacent. Let i(G) denote the size of a smallest maximal independent set and let fl(G) denote the size of a largest independent set. Equivalently, i(G) is the size of a smallest independent dominating set. The number i(G) is called the independent dominating number and fl(G) is called the independence number. A set S C_ V is a dominating set if N[S] = V. In other words, every vertex in V is either in S or adjacent to a vertex of S. Let y(G) and F(G) denote the sizes of smallest and largest minimal dominating sets of a graph G, respectively. The number ),(G) is called the domination number and F(G) is called the upper domination number. Note that any maximal independent set is a dominating set. For a dominating set S to be minimal, each vertex x c S must have a private neighbour, otherwise the smaller set S {x} is dominating. A set S is irredundant if for all v E S, v has a private neighbour with respect to S. That is, for all v E S, N[v] N [ S {v}] ~ 0. Any minimal dominating set is therefore irredundant. Moreover, an irredundant set which is dominating is a minimal dominating set. Let ir(G) and IR(G) denote the sizes of smallest and largest maximal irredundant sets of a graph G, respectively. The number ir(G) is called the irredundance number and IR(G) is called the upper irredundance number. Cockayne et al. [3] proved the following inequality: Theorem 1 (Cockayne [3]). For any graph G, ir(G) ~< y(G) ~< i(G) <~ fl(G) <~ F(G) <<. IR(G). The parameters ir(G), 7(G) and i(G) are collectively known as the lower domination parameters. The parameters fl(G), F(G) and IR(G) are known as the upper domination parameters. A classical theorem in graph theory is due to Gallai [4]. Here, ~0(G) is the vertex covering number, the smallest size of a set of vertices needed so that every edge has at least one end vertex in the set. Theorem 2 (Gallai [4]). For any graph G, fl(G) + ct0(G) = n. A spanning forest of a graph G is a spanning subgraph which contains no cycles. Let e(G) denote the maximum number of pendant edges in a spanning forest of G. In [6], Nieminen proved the following: Theorem 3 (Nieminen [6]). For any nontrivial connected graph G, y(G) + e(G) = n. A Gallai-type Theorem has the form x(G)+ y ( G ) : n where x(G), y(G) are parameters defined on the graph G. In [2], Cockayne et al. survey Gallai-type theorems. In G.S. Domke/Discrete Mathematics 167/168 (1997) 237-248 239 this spirit, we will investigate the lower domination parameters and combine them with the maximum degree, then look at the upper domination parameters combined with the minimum degree. 2. Graphs which satisfy i( G) + A( G) = n The first parameter we will consider is i(G), the independent domination number. Theorem 4. For any graph G, i(G) + A(G) <% n. Proof. Let x be a vertex of degree A(G). Let the set S be the vertex x together with any independent dominating set of V N[x]. Then S will independently dominate the graph and i(G) + A(G) <~ IS[ + A(G) <,% n. [] From this theorem and the inequality in Theorem 1, we get the following corollary: Corollary 1. For any graph G, ?(G) + A(G) % n and ir(G) + A(G) <~ n. For any inequality, it is interesting to discover conditions which guarantee equality. In this section, we are interested in finding those graphs which have i( G) + A( G) = n. Examples of graphs for which i( G) + A( G) = n include Kn and any graph with A ( G ) = n 1 or A ( G ) = n 2. Theorem 5. Let G be a graph with i(G) + A(G) = n and let x be a vertex o f degree A(G). Then V N[x] is an independent set. Proof. Suppose there is an edge in V N[x]. Any independent dominating set of V N [ x ] will have size at most I V N [ x ] [ 1. Thus i(G) <~ IV N [ x ] l 1 + I{x}[ = n A ( G ) l