Lisa R. Markus

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.

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.

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.