Abstract:
For a fixed integer
b>1
b>1
, a set
D⊆V
D⊆V
is called a b-disjunctive 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 b-Disjunctive Domination Problem (MbDDP) is to find a b-disjunctive dominating set of minimum cardinality. The cardinality of a minimum b-disjunctive dominating set of G is called the b-disjunctive domination number of G, and is denoted by
γ
d
b
(G)
γbd(G)
. Given a positive integer k and a graph G, the b-Disjunctive Domination Decision Problem (bDDDP) is to decide whether G has a b-disjunctive 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 b-disjunctive dominating set of a proper interval graph for
b=2
b=2
. Next we tighten the NP-completeness of bDDDP by showing that it remains NP-complete 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(loglog|V|)
)
(|V|O(loglog|V|))
. Finally, we show that MbDDP is APX-complete for bipartite graphs with maximum degree
max{b,4}
max{b,4}
.
