Complexity of distance paired-domination problem in graphs

Suppose G = (V, E) is a simple graph and k is a fixed positive integer. A subset D subset of V is a distance k-dominating set of G if for every u is an element of V. there exists a vertex v is an element of D such that d(G)(u, v) <= k, where d(G)(u, v) is the distance between u and v in G. A set D subset of V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph GD] contains a perfect matching. Given a graph G = (V, E) and a fixed integer k > 0, the MIN DISTANCE k-PAIRED-DOM SET problem is to find a minimum cardinality distance k-paired-dominating set of G. In this paper, we show that the decision version of MIN DISTANCE k-PAIRED-DOM SET iS NP-complete for undirected path graphs. This strengthens the complexity of decision version Of MIN DISTANCE k-PAIRED-DOM SET problem in chordal graphs. We show that for a given graph G, unless NP subset of DTIME (n(0)((log) (log) (n)) MIN DISTANCE k-PAIRED-Dom SET problem cannot be approximated within a factor of (1 -epsilon ) In n for any epsilon > 0, where n is the number of vertices in G. We also show that MIN DISTANCE k-PAIRED-DOM SET problem is APX-complete for graphs with degree bounded by 3. On the positive side, we present a linear time algorithm to compute the minimum cardinality of a distance k-paired-dominating set of a strongly chordal graph G if a strong elimination ordering of G is provided. We show that for a given graph G, MIN DISTANCE k-PAIRED-DOM SET problem can be approximated with an approximation factor of 1 + In 2 + k . In(Delta(G)), where Delta(G) denotes the maximum degree of G. (C) 2012 Elsevier B.V All rights reserved.

Hardness results for computing optimal locally Gabriel graphs

Delaunay and Gabriel graphs are widely studied geo-metric proximity structures. Motivated by applications in wireless routing, relaxed versions of these graphs known as Locally Delaunay Graphs (LDGs) and Lo-cally Gabriel Graphs (LGGs) have been proposed. We propose another generalization of LGGs called Gener-alized Locally Gabriel Graphs (GLGGs) in the context when certain edges are forbidden in the graph. Unlike a Gabriel Graph, there is no unique LGG or GLGG for a given point set because no edge is necessarily in-cluded or excluded. This property allows us to choose an LGG/GLGG that optimizes a parameter of interest in the graph. We show that computing an edge max-imum GLGG for a given problem instance is NP-hard and also APX-hard. We also show that computing an LGG on a given point set with dilation ≤k is NP-hard. Finally, we give an algorithm to verify whether a given geometric graph G= (V, E) is a valid LGG.