Domination in graphs with bounded propagation: algorithms, formulations and hardness results

Abstract

We introduce a hierarchy of problems between the Dominating Set problem and the Power Dominating Set (PDS) problem called the ℓ-round power dominating set (ℓ-round PDS, for short) problem. For ℓ = 1, this is the Dominating Set problem, and for ℓ ≥ n − 1, this is the PDS problem; here n denotes the number of nodes in the input graph. In PDS the goal is to find a minimum size set of nodes S that power dominates all the nodes, where a node v is power dominated if (1) v is in S or it has a neighbor in S, or (2) v has a neighbor u such that u and all of its neighbors except v are power dominated. Note that rule (1) is the same as for the Dominating Set problem, and that rule (2) is a type of propagation rule that applies iteratively. The ℓ-round PDS problem has the same set of rules as PDS, except we apply rule (2) in "parallel" in at most ℓ − 1 rounds. We prove that ℓ-round PDS cannot be approximated better than 2 log 1−ǫ n even for ℓ = 4 in general graphs. We provide a dynamic programming algorithm to solve ℓ-round PDS optimally in polynomial time on graphs of bounded tree-width. We present a PTAS (polynomial time approximation scheme) for ℓ-round PDS on planar graphs for ℓ = O( log n log log n ). Finally, we give integer programming formulations for ℓ-round PDS.

Power Domination in Circular-Arc Graphs

“…For ease of reference, the vertices in V i \ V i−1 , i > 0 are called the i th generation descendants (i-descendants for short) of those in V 0 . Note that the Induced Observation Rules are equivalent to the original observation rules [1,2,8,15,21,31]. In addition, given a graph G = (V, E), the observed graph of some kernel V 0 can be computed in O(|V | + |E|) time by the Induced Observation Rules [8].…”

“…, a n }. For two successive right endpoints b i and b i+1 , 1 Aazami and Stilp [1] independently presented a similar extension, called the -round PDS problem.…”

Generalized Power Domination: Propagation Radius and Sierpiński Graphs

“…Actually, several hundreds of measurement units for monitoring an electric power system have been installed world wide. For a detailed description of the application in power networks and for related references see the very instructive introduction of [1].…”

“…For complexity results and related topics (like parametrized power domination complexity and approximation algorithms) see [1,2,4,12,13,21]. In particular, the problem of deciding whether there exists a power dominating set of a given size is NP-complete for planar bipartite graphs [4].…”

“…In the subsequent section we introduce the concept of a propagation radius as the minimum number of propagation steps over all kPD-sets. We were in particular motivated by the investigations of Aazami in [1], where the problem of determining a minimum size power dominating set is studied under the additional condition that the number of propagation steps in bounded by a fixed constant. In the final section the propagation radius of Sierpiński graphs S n p is determined for most of the parameters p and n.…”