For a graph G=(V,E), a set D⊆V is called a \emph{disjunctive
dominating set} of G if for every vertex v∈V∖D, v is either
adjacent to a vertex of D or has at least two vertices in D at distance 2
from it. The cardinality of a minimum disjunctive dominating set of G is
called the \emph{disjunctive domination number} of graph G, and is denoted by
γ2d(G). The \textsc{Minimum Disjunctive Domination Problem} (MDDP)
is to find a disjunctive dominating set of cardinality γ2d(G).
Given a positive integer k and a graph G, the \textsc{Disjunctive
Domination Decision Problem} (DDDP) is to decide whether G has a disjunctive
dominating set of cardinality at most k. In this article, we first propose a
linear time algorithm for MDDP in proper interval graphs. Next we tighten the
NP-completeness of DDDP by showing that it remains NP-complete even in chordal
graphs. We also propose a (ln(Δ2+Δ+2)+1)-approximation
algorithm for MDDP in general graphs and prove that MDDP can not be
approximated within (1−ϵ)ln(∣V∣) for any ϵ>0 unless NP
⊆ DTIME(∣V∣O(loglog∣V∣)). Finally, we show that MDDP is
APX-complete for bipartite graphs with maximum degree 3