(1, J)-set Problem in Graphs

A subset D ź V of a graph G = ( V , E ) is a ( 1 , j ) -set (Chellali etźal., 2013) if every vertex v ź V ź D is adjacent to at least 1 but not more than j vertices in D . The cardinality of a minimum ( 1 , j ) -set of G , denoted as γ ( 1 , j ) ( G ) , is called the ( 1 , j ) -domination number of G . In this paper, using probabilistic methods, we obtain an upper bound on γ ( 1 , j ) ( G ) for j ź O ( log Δ ) , where Δ is the maximum degree of the graph. The proof of this upper bound yields a randomized linear time algorithm. We show that the associated decision problem is NP-complete for choral graphs but, answering a question of Chellali etźal., provide a linear-time algorithm for trees for a fixed j . Apart from this, we design a polynomial time algorithm for finding γ ( 1 , j ) ( G ) for a fixed j in a split graph, and show that ( 1 , j ) -set problem is fixed parameter tractable in bounded genus graphs and bounded treewidth graphs.