Gayla S. Domke

Restrained domination in graphs

Abstract In this paper, we initiate the study of a variation of standard domination, namely restrained domination. Let G =( V , E ) be a graph. A restrained dominating set is a set S ⊆ V where every vertex in V − S is adjacent to a vertex in S as well as another vertex in V − S . The restrained domination number of G , denoted by γ r ( G ), is the smallest cardinality of a restrained dominating set of G . We determine best possible upper and lower bounds for γ r ( G ), characterize those graphs achieving these bounds and find best possible upper and lower bounds for γ r (G)+γ r ( G ) where G is a connected graph. Finally, we give a linear algorithm for determining γ r ( T ) for any tree and show that the decision problem for γ r ( G ) is NP-complete even for bipartite and chordal graphs.

Restrained domination in trees

Abstract Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V−S . The restrained domination number of G , denoted by γ r (G) , is the smallest cardinality of a restrained dominating set of G . We show that if T is a tree of order n , then γ r (T)⩾⌈(n+2)/3⌉ . Moreover, we constructively characterize the extremal trees T of order n achieving this lower bound.

Maximum sizes of graphs with given domination parameters

We find the maximum number of edges for a graph of given order and value of parameter for several domination parameters. In particular, we consider the total domination and independent domination numbers.

Fractional domination of strong direct products

Let G be a graph with edges E(G). A subset of the nodes dominates G if each node of G is either in or is adjacent to a member of the subset. The domination number of G, y(G), is the minimum size of a dominating set. In 1963, Vizing [3] conjectured that for all graphs G and H, y(G 0 H) 3 y(G)?(H) w h ere G @ H is the Cartesian product of G and H. We prove an analogous result for the fractional domination number. We can redefine y(G) as the value of the integer programming problem. For n-vectors x and y, let x 2 y (X < y) mean Xi > yi (xi < yi) for all i. Let 1, and 0, be the n-vectors whose components are all one or all zero, respectively. If N(G) is the neighborhood matrix of G (the adjacency matrix plus the identity matrix), then

Gallai-type theorems and domination parameters

Abstract Let γ ( G ) denote the minimum cardinality of a dominating set of a graph G = ( V , E ). A longstanding upper bound for γ ( G ) is attributed to Berge: For any graph G with n vertices and maximum degree Δ ( G ), γ ( G ) ⩽ n − Δ ( G ). We characterise connected bipartite graphs which achieve this upper bound. For an arbitrary graph G we furnish two conditions which are necessary if γ ( G ) + Δ ( G ) = n and are sufficient to achieve n − 1 ⩽ γ ( G ) + Δ ( 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 − Δ ( 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 δ ( G ), IR ( G ) ⩽ n - δ ( 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.

The bondage and reinforcement numbers of g f for some graphs

Abstract For any graph G , a real-valued function g : V ( G ) → [0, 1] is called a dominating function if for every v ϵ V ( G ), Σ wϵN [ v ] g ( w ) ⩾ 1 The fractional domination number is defined to be γ f ( G ) =min{ Σ vϵV ( G ) g ( v ): g is a dominating function of G }. In this paper, we initiate the study of bondage and reinforcement associated with fractional domination. The bondage number of γ f , b f ( G ), is defined to be the minimum cardinality of a set of edges whose removal from G results in a graph G 1 satisfying γ f ( G 1 ) > γ f ( G ). The reinforcement number of γ f , r f ( G ), is defined to be the minimum cardinality of a set of edges which when added to G results in a graph G ″ satisfying γ f ( G ″) γ f ( G ). We will give exact values of b f ( G ) and r f ( G ) for some classes of graphs.

The rank of a graph after vertex addition

Abstract Let r(G) denote the rank of the adjacency matrix of a graph G. When a vertex and its incident edges are deleted from G, the rank of the resultant graph cannot exceed r(G) and can decrease by at most 2. The problem of determining the exact effect of adding a single vertex to a graph is more difficult, since the number of edges that can be added with this vertex is variable. The rank of the new graph cannot decrease and it can increase by at most 2. We obtain results examining several cases of vertex addition.

On parameters related to strong and weak domination in graphs

Abstract Let G be a graph. Then μ ( G )⩽| V ( G )|− δ ( G ) where μ ( G ) denotes the weak or independent weak domination number of G and μ ( G )⩽| V ( G )|− Δ ( G ) where μ ( G ) denotes the strong or independent strong domination number of G . We give necessary and sufficient conditions for equality to hold in each case and also describe specific classes of graphs for which equality holds. Finally, we show that the problems of computing i w and i st are NP-hard, even for bipartite graphs.