Abstract
A subset D⊆V of a graph G=(V,E) is a (1,j)-set (Chellali et al., 2013) if every vertex v∈V∖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≥O(logΔ), 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 et al., 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.
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