Unique Response Roman Domination: Complexity and Algorithms
A function f : V ( G ) → { 0 , 1 , 2 } is called a Roman dominating function on G = ( V ( G ) , E ( G ) ) if for every vertex v with f ( v ) = 0 , there exists a vertex u ∈ N G ( v ) such that f ( u ) = 2 . A function f : V ( G ) → { 0 , 1 , 2 } induces an ordered partition ( V 0 , V 1 , V 2 ) of V...
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| Published in: | Algorithmica Vol. 85; no. 12; pp. 3889 - 3927 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.12.2023
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online Access: | Get full text |
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| Summary: | A function
f
:
V
(
G
)
→
{
0
,
1
,
2
}
is called a
Roman dominating function
on
G
=
(
V
(
G
)
,
E
(
G
)
)
if for every vertex
v
with
f
(
v
)
=
0
, there exists a vertex
u
∈
N
G
(
v
)
such that
f
(
u
)
=
2
. A function
f
:
V
(
G
)
→
{
0
,
1
,
2
}
induces an ordered partition
(
V
0
,
V
1
,
V
2
)
of
V
(
G
), where
V
i
=
{
v
∈
V
(
G
)
:
f
(
v
)
=
i
}
for
i
∈
{
0
,
1
,
2
}
. A function
f
:
V
(
G
)
→
{
0
,
1
,
2
}
with ordered partition
(
V
0
,
V
1
,
V
2
)
is called a
unique response Roman function
if for every vertex
v
with
f
(
v
)
=
0
,
|
N
G
(
v
)
∩
V
2
|
≤
1
, and for every vertex
v
with
f
(
v
)
=
1
or 2,
|
N
G
(
v
)
∩
V
2
|
=
0
. A function
f
:
V
(
G
)
→
{
0
,
1
,
2
}
is called a
unique response Roman dominating function
(URRDF) on
G
if it is a unique response Roman function as well as a Roman dominating function on
G
. The weight of a unique response Roman dominating function
f
is the sum
f
(
V
(
G
)
)
=
∑
v
∈
V
(
G
)
f
(
v
)
, and the minimum weight of a unique response Roman dominating function on
G
is called the
unique response Roman domination number
of
G
and is denoted by
u
R
(
G
)
. Given a graph
G
, the
Min-URRDF
problem asks to find a unique response Roman dominating function of minimum weight on
G
. In this paper, we study the algorithmic aspects of
Min-URRDF
. We show that the decision version of
Min-URRDF
remains NP-complete for chordal graphs and bipartite graphs. We show that for a given graph with
n
vertices,
Min-URRDF
cannot be approximated within a ratio of
n
1
-
ε
for any
ε
>
0
unless
P
=
NP
. We also show that
Min-URRDF
can be approximated within a factor of
Δ
+
1
for graphs having maximum degree
Δ
. On the positive side, we design a linear-time algorithm to solve
Min-URRDF
for distance-hereditary graphs. Also, we show that
Min-URRDF
is polynomial-time solvable for interval graphs, and strengthen the result by showing that
Min-URRDF
can be solved in linear-time for proper interval graphs, a proper subfamily of interval graphs. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-023-01171-7 |