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 |
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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∈NG(v) such that f(u)=2. A function f:V(G)→{0,1,2} induces an ordered partition (V0,V1,V2) of V(G), where Vi={v∈V(G):f(v)=i} for i∈{0,1,2}. A function f:V(G)→{0,1,2} with ordered partition (V0,V1,V2) is called a unique response Roman function if for every vertex v with f(v)=0, |NG(v)∩V2|≤1, and for every vertex v with f(v)=1 or 2, |NG(v)∩V2|=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 uR(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 n1-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. |
| Author | Chaudhary, Juhi Banerjee, Sumanta Pradhan, Dinabandhu |
| Author_xml | – sequence: 1 givenname: Sumanta surname: Banerjee fullname: Banerjee, Sumanta organization: Department of Mathematics and Computing, Indian Institute of Technology (ISM) – sequence: 2 givenname: Juhi surname: Chaudhary fullname: Chaudhary, Juhi organization: Department of Computer Science, Ben-Gurion University of the Negev – sequence: 3 givenname: Dinabandhu surname: Pradhan fullname: Pradhan, Dinabandhu email: dina@iitism.ac.in organization: Department of Mathematics and Computing, Indian Institute of Technology (ISM) |
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| Cites_doi | 10.1016/j.dam.2016.09.035 10.1007/s10878-012-9500-0 10.1016/0020-0190(84)90126-1 10.1016/j.dam.2016.03.017 10.1007/s00453-020-00705-7 10.1016/j.tcs.2019.08.017 10.1145/2591796.2591884 10.1016/j.dam.2021.08.020 10.1080/00029890.2000.12005243 10.1287/moor.4.3.233 10.1016/j.disc.2003.06.004 10.1016/j.dam.2019.01.038 10.1007/3-540-63890-3_37 10.1007/s12190-020-01345-4 10.1137/1.9780898719796 10.1016/j.dam.2022.09.017 10.1016/0020-0190(88)90091-9 10.1038/scientificamerican1299-136 10.1016/0196-6774(85)90001-X 10.1080/00207160.2017.1301437 10.1007/978-3-642-58412-1 10.1145/800157.805047 10.1002/net.1 10.1016/j.dam.2008.01.011 10.1016/j.ipl.2018.01.004 10.1016/j.disc.2007.03.020 10.7151/dmgt.2067 10.1016/j.dam.2011.03.013 10.1016/j.tcs.2022.02.006 10.1137/S0097539792238431 10.1016/S0166-218X(98)00060-2 10.1016/0166-218X(90)90131-U 10.1016/j.dam.2017.10.027 10.1016/S0020-0190(03)00298-9 10.1016/S0166-218X(97)00125-X 10.1137/0217032 10.1016/j.dam.2005.07.011 10.1016/j.dam.2015.11.013 10.1016/0304-3975(87)90067-3 10.1016/j.akcej.2020.01.005 10.1007/s10878-019-00457-3 10.1016/0095-8956(86)90043-2 |
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| Issue | 12 |
| Keywords | Domination Roman domination Polynomial-time algorithm Unique response Roman domination Unique response Roman function NP-completeness |
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| Snippet | 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 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∈NG(v) such that... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Apexes Approximation Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphs Mathematical functions Mathematics of Computing Minimum weight Polynomials Theory of Computation |
| Title | Unique Response Roman Domination: Complexity and Algorithms |
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