Article
Kyungpook Mathematical Journal 2021; 61(3): 661669
Published online September 30, 2021
Copyright © Kyungpook Mathematical Journal.
Complexity Issues of Perfect Roman Domination in Graphs
Padamutham Chakradhar^{*}, Palagiri Venkata Subba Reddy
Department of Computer Science and Engineering, National Institute of Techonology, Warangal, India
email : corneliusp7@gmail.com and venkatpalagiri@gmail.com
Received: November 6, 2019; Revised: October 12, 2021; Accepted: November 23, 2020
Abstract
For a simple, undirected graph G = (V, E), a perfect Roman dominating function (PRDF) f : V → {0, 1, 2} has the property that, every vertex u with f(u) = 0 is adjacent to xactly one vertex v for which f(v) = 2. The weight of a PRDF is the sum
Keywords: Roman domination, Perfect Roman domination, NPcomplete.
1. Introduction
Let
A vertex
Roman domination has been introduced by Cockayne et al. in [4]. A function
The concept of perfect Roman domination was introduced in 2018 by Henning et al. in [8]. A function
Given a graph
2. Complexity Results
In this section, we show that the PRDF problem is NPcomplete for star convex bipartite graphs and comb convex bipartite graphs by giving a polynomial time reduction from a wellknown NPcomplete problem, Exact3Cover
EXACT3COVER (X3C)
The decision version of perfect Roman dominating function problem is defined as follows.
PERFECT ROMAN DOMINATING FUNCTION PROBLEM (PRDFP)
Create vertices

Figure 1. Star Graph

Figure 2. Construction of a star convex bipartite graph from an instance of X3C
Suppose
Clearly,
Conversely, suppose that
Since each
Create vertices

Figure 3. Comb Graph

Figure 4. Construction of a comb convex bipartite graph from an instance of X3C
Suppose
Clearly,
Conversely, suppose that
Now, the following result is immediate from Theorem 2.1 and Theorem 2.2.
3. Threshold Graphs
In this section, we determine the perfect Roman domination number of threshold graph.
Definition 3.1. A graph
Although several characterizations defined for threshold graphs, We use the following characterization of threshold graphs given in [11] to prove that the perfect Roman domination number can be computed in linear time for threshold graphs.
A graph
Theorem 3.2. Let
Clearly,
Now, the following result is immediate from Theorem 3.1.
Theorem 3.3. PRDF problem can be solvable in linear time for threshold graphs.
4. Chain Graphs
In this section, we propose a method to compute the perfect Roman domination number of a chain graph in linear time. A bipartite graph
Proposition 4.1. Let
(a) If
(b) If
(c) If
If
Theorem 4.2. Let
Clearly,
Clearly,
Since
If the chain graph
Theorem 4.3. PRDF problem can be solvable in linear time for chain graphs.
5. Bounded Treewidth Graphs
Let
Theorem 5.1.([
Theorem 5.2. Given a graph
where
Now, the following result is immediate from Theorem 5.1 and Theorem 5.2.
Theorem 5.3. PRDF problem can be solvable in linear time for bounded treewidth graphs.
6. Conclusion
In this paper, we have shown that the decision problem associated with
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