Abstract:
In a graph G = (V , E) without an isolated vertex, a dominating set D ⊆ V is a paired
dominating set if the graph G[D] induced by D has a perfect matching. Further, a set
D ⊆ V is a disjunctive dominating set of G if for each vertex v ∈ V , either N G [v] ∩ D = ∅
or there are at least two vertices in D whose distance from v is two in G. We introduce
the notion of paired disjunctive domination in graphs. A disjunctive dominating set D ⊆ V
in the graph G is a paired disjunctive dominating set if G[D] has a perfect matching. The
minimum cardinality of a paired disjunctive dominating set of G is the paired disjunctive
domination number, denoted by γ d
pr(G).
In this article, we compute the exact value of γ d
pr(G) when G is a path, cycle, cograph,
chain graph, and split graph. We prove that the decision version of the problem is NPcomplete for planar graphs, bipartite graphs, and chordal graphs and design a polynomialtime algorithm to compute a minimum cardinality paired disjunctive dominating set in
interval graphs. Further, we obtain lower and upper bounds on the approximation ratio of
the problem and proved that the problem is APX-complete for the graphs with maximum
degree 4.
