An efficient algorithm to solve the distancek-domination problem on permutation graphs

Abstract

Let G =(V, E) be a graph and k be a fixed positive integer. A distance k-dominating set in a graph G, is a set of vertices D in V such that every vertex in V\D is at distance at most k from some vertex in D. The minimum cardinality distance k-dominating set in G is the distance k-domination number γ k . The distance k-domination problem is to find a γ k in G. This problem generalizes the dominating set problem, a central problem in theoretical computer science and is therefore NP-complete for general graphs. The main result of this paper is a better time bound for distance k-domination on permutation graphs by exploiting the structure of permutation in a direct way.

Generalized distance domination problems and their complexity on graphs of bounded mim-width

“…For instance, blindly applying our results to solve Distance-r Dominating Set on permutation graphs yields an algorithm that runs in time O(n 8 ): Permutation graphs have linear mim-width 1 (with a corresponding decomposition tree that can be computed in linear time) [3, Lemmas 2 and 5], so we can apply Corollary 7(i). However, there is an algorithm that solves Distance-r Dominating Set on permutation graphs in time O(n 2 ) [16]; a much faster runtime. A concrete example of improving a mim-width based algorithm on a specific graph class has recently been provided by Chiarelli et al [8] who gave algorithms for the (Total) k-Dominating Set problems that run in time O(n 3k ) on proper interval graphs.…”

Computation of minimum d-hop connected dominating set of trees in O(n) time

“…Natarajan et al [26] have done some fundamental works on hop domination number on some special class of graphs. Also, on permutation-graphs Rana et al [29] set up an effective algorithm to find a distance-k dominating set. Later, Ayyaswamy et al [3] worked on the upper and lower limits of the hop D-number on trees.…”