Power Domination in Circular-Arc Graphs

Abstract

A set S ⊆ V is a power dominating set (PDS) of a graph G = (V, E) if every vertex and every edge in G can be observed based on the observation rules of power system monitoring. The power domination problem involves minimizing the cardinality of a PDS of a graph. We consider this combinatorial optimization problem and present a linear time algorithm for finding the minimum PDS of an interval graph if the interval ordering of the graph is provided. In addition, we show that the algorithm, which runs in Θ(n log n) time, where n is the number of intervals, is asymptotically optimal if the interval ordering is not given. We also show that the results hold for the class of circular-arc graphs.

New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

“…k-domination and total k-domination problems were also studied with respect to their (in)approximability properties, both in general [18] and in restricted graph classes [2], as well as from the parameterized complexity point of view [9,37]. Besides k-domination and total k-domination, other variants of domination problems solvable in polynomial time in the class of proper interval graphs (or in some of its superclasses) include k-tuple domination for all k ≥ 1 [48] (see also [47] and, for k = 2, [57]), connected domination [59], independent domination [23], paired domination [15], efficient domination [11], liar's domination [54], restrained domination [55], eternal domination [5], power domination [49], outer-connected domination [51], Roman domination [50], Grundy domination [7], etc.…”

Power domination in certain chemical structures

“…The power domination problem has efficient polynomial time algorithms for the classes of trees [13], graphs with bounded treewidth [12], block graphs [24], block-cactus graphs [14], interval graphs [16], grids [20], honeycomb meshes [23] and circular-arc graphs [17]. Upper bounds on the power domination number are given for a connected graph with at least three vertices, for a connected claw-free cubic graph [25], for hypercubes [5], and for generalized Petersen graphs [3].…”

$$k$$ k -Power domination in block graphs

“…From the algorithmic point of view, the power domination problem was known to be NPcomplete even when restricted to bipartite, planar, circle and split graphs (Guo et al 2008;Haynes et al 2002) and approximation algorithms were given in Aazami and Stilp (2009). On the other hand, polynomial time algorithms for the power domination problem were given for trees (Chang et al 2012;Guo et al 2008;Haynes et al 2002), for interval graphs (Liao and Lee 2005), for block graphs (Xu et al 2006) and for circular-arc graphs (Liao and Lee 2013). Parameterized results were given in Kneis et al (2006).…”