Abstract:
For a fixed integer
b>1
b>1
, a set
D⊆V
D⊆V
is called a bdisjunctive dominating set of the graph
G=(V,E)
G=(V,E)
if for every vertex
v∈V∖D
v∈V∖D
, v is either adjacent to a vertex of D or has at least b vertices in D at distance 2 from it. The Minimum bDisjunctive Domination Problem (MbDDP) is to find a bdisjunctive dominating set of minimum cardinality. The cardinality of a minimum bdisjunctive dominating set of G is called the bdisjunctive domination number of G, and is denoted by
γ
d
b
(G)
γbd(G)
. Given a positive integer k and a graph G, the bDisjunctive Domination Decision Problem (bDDDP) is to decide whether G has a bdisjunctive dominating set of cardinality at most k. In this paper, we first show that for a proper interval graph G,
γ
d
b
(G)
γbd(G)
is equal to
γ(G)
γ(G)
, the domination number of G for
b≥3
b≥3
and observe that
γ
d
b
(G)
γbd(G)
need not be equal to
γ(G)
γ(G)
for
b=2
b=2
. We then propose a polynomial time algorithm to compute a minimum cardinality bdisjunctive dominating set of a proper interval graph for
b=2
b=2
. Next we tighten the NPcompleteness of bDDDP by showing that it remains NPcomplete even in chordal graphs. We also propose a
(ln(
Δ
2
+(b−1)Δ+b)+1)
(ln(Δ2+(b−1)Δ+b)+1)
approximation algorithm for MbDDP, where
Δ
Δ
is the maximum degree of input graph
G=(V,E)
G=(V,E)
and prove that MbDDP cannot be approximated within
(1−ϵ)ln(V)
(1−ϵ)ln(V)
for any
ϵ>0
ϵ>0
unless NP
⊆
⊆
DTIME
(V

O(loglogV)
)
(VO(loglogV))
. Finally, we show that MbDDP is APXcomplete for bipartite graphs with maximum degree
max{b,4}
max{b,4}
.