An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework for solving the weighted efficient domination problems based on a reduction to the maximum weight independent set problem in the square of the input graph. Using this approach, we improve on several previous results from the literature by deriving polynomial-time algorithms for the weighted efficient domination problems in the classes of dually chordal and AT-free graphs. In particular, this answers a question by Lu and Tang regarding the complexity of the minimum weight efficient domination problem in strongly chordal graphs.
For a graph G = (V, E) the minimum line-distortion problem asks for the minimum k such that there is a mapping f of the vertices into points of the line such that for each pair of vertices x, y the distance on the line |f (x) − f (y)| can be bounded by the term dG(x, y) ≤ |f (x) − f (y)| ≤ k dG(x, y), where dG(x, y) is the distance in the graph. The minimum bandwidth problem minimizes the term maxuv∈E |f (u) − f (v)|, where f is a mapping of the vertices of G into the integers {1, . . . , n}. We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show:if a graph G can be embedded into the line with distortion k, then G admits a Robertson-Seymour's path-decomposition with bags of diameter at most k in G; -for every class of graphs with path-length bounded by a constant, there exist an efficient constantfactor approximation algorithm for the minimum line-distortion problem and an efficient constantfactor approximation algorithm for the minimum bandwidth problem; -there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph; -AT-free graphs and some intersection families of graphs have path-length at most 2; -for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum linedistortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem.
Let G = (V, E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v ∈ V dominates its closed neighborhoodThe ED problem (EED problem, respectively) asks for the existence of an e.d. set (e.e.d. set, respectively) in the given graph. We give a unified framework for investigating the complexity of these problems on various classes of graphs. In particular, we solve some open problems and give linear time algorithms for ED and EED on dually chordal graphs. We extend the two problems to hypergraphs and show that ED remains NP-complete on α-acyclic hypergraphs, and is solvable in polynomial time on hypertrees, while EED is polynomial on α-acyclic hypergraphs and NP-complete on hypertrees.
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