Linear-Time Algorithm for the Paired-Domination Problem in Convex Bipartite Graphs

2011 ◽  
Vol 50 (4) ◽  
pp. 721-738 ◽  
Author(s):  
Ruo-Wei Hung
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Paired Domination ◽  
Convex Bipartite Graphs

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

scholarly journals Complexity issues of perfect secure domination in graphs

2021 ◽  
Vol 55 ◽  
pp. 11
Author(s):  
P. Chakradhar ◽  
P. Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Split Graphs ◽  
Secure Domination ◽  
Convex Bipartite Graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

A linear time algorithm to compute a minimum restrained dominating set in proper interval graphs

2015 ◽  
Vol 07 (02) ◽  
pp. 1550020 ◽  
Author(s):  
B. S. Panda ◽  
D. Pradhan
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Time Algorithm ◽  
Interval Graph ◽  
Chordal Graphs ◽  
Linear Time Algorithm ◽  
Circle Graphs ◽  
Restrained Domination ◽  
Proper Interval Graph ◽  
Domination Problem

A set D ⊆ V is a restrained dominating set of a graph G = (V, E) if every vertex in V\D is adjacent to a vertex in D and a vertex in V\D. Given a graph G and a positive integer k, the restrained domination problem is to check whether G has a restrained dominating set of size at most k. The restrained domination problem is known to be NP-complete even for chordal graphs. In this paper, we propose a linear time algorithm to compute a minimum restrained dominating set of a proper interval graph. We present a polynomial time reduction that proves the NP-completeness of the restrained domination problem for undirected path graphs, chordal bipartite graphs, circle graphs, and planar graphs.

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