Algorithmic Aspects of Semitotal Domination in Graphs

11/29/2017
by   Michael A. Henning, et al.
0

For a graph G=(V,E), a set D ⊆ V is called a semitotal dominating set of G if D is a dominating set of G, and every vertex in D is within distance 2 of another vertex of D. The Minimum Semitotal Domination problem is to find a semitotal dominating set of minimum cardinality. Given a graph G and a positive integer k, the Semitotal Domination Decision problem is to decide whether G has a semitotal dominating set of cardinality at most k. The Semitotal Domination Decision problem is known to be NP-complete for general graphs. In this paper, we show that the Semitotal Domination Decision problem remains NP-complete for planar graphs, split graphs and chordal bipartite graphs. We give a polynomial time algorithm to solve the Minimum Semitotal Domination problem in interval graphs. We show that the Minimum Semitotal Domination problem in a graph with maximum degree Δ admits an approximation algorithm that achieves the approximation ratio of 2+3(Δ+1), showing that the problem is in the class log-APX. We also show that the Minimum Semitotal Domination problem cannot be approximated within (1 - ϵ) |V| for any ϵ > 0 unless NP ⊆ DTIME (|V|^O( |V|)). Finally, we prove that the Minimum Semitotal Domination problem is APX-complete for bipartite graphs with maximum degree 4.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/01/2019

Complexity and Algorithms for Semipaired Domination in Graphs

For a graph G=(V,E) with no isolated vertices, a set D⊆ V is called a se...
research
10/14/2019

Algorithm and hardness results on neighborhood total domination in graphs

A set D⊆ V of a graph G=(V,E) is called a neighborhood total dominating ...
research
10/30/2018

Simultaneous Dominating Set for Spanning Tree Factorings

For a connected graph G we call a set F a spanning tree factoring of G i...
research
08/01/2023

Minimum k-Hop Dominating Sets in Grid Graphs

Given a graph G, the k-hop dominating set problem asks for a vertex subs...
research
12/16/2020

Maximum 0-1 Timed Matching on Temporal Graphs

Temporal graphs are graphs where the topology and/or other properties of...
research
10/30/2020

Approximability results for the p-centdian and the converse centdian problems

Given an undirected graph G=(V,E,l) with a nonnegative edge length funct...
research
05/15/2019

Perfect Italian domination on planar and regular graphs

A perfect Italian dominating function of a graph G=(V,E) is a function f...

Please sign up or login with your details

Forgot password? Click here to reset