Generalized Liar's Dominating Set in Graphs
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In this article, we study generalized liar's dominating set problem in graphs. Let G=(V,E) be a simple undirected graph. The generalized liar's dominating set, called as the distance-d(m,ℓ)-liar's dominating set, is a subset L⊆ V such that (i) each vertex in V is distance-d dominated by at least m vertices in L, and (ii) each pair of distinct vertices in V is distance-d dominated by at least ℓ vertices in L, where m,ℓ,d are positive integers and m < ℓ. Here, a vertex v is distance-d dominated by another vertex u means the shortest path distance between u and v is at most d in G. We first consider distance-1 (m,ℓ)-liar's dominating set problem and prove that it is NP-complete. Next, we consider distance-d(m,ℓ)-liar's dominating set problem and show that it is also NP-complete. These liar's dominating set problems are generalized version of liar's dominating set problem as researcher studied only distance-1(2,3)-liar's dominating set problem in literature. We also prove that (i) distance-1 (m,ℓ)-liar's dominating set problem cannot be approximated within a factor of (1/2- ε)ln |V| for any ε>0, unless NP ⊆ DTIME(|V|^O(loglog|V|)), and (ii) distance-d(m,ℓ)-liar's dominating set problem cannot be approximated within a factor of (1/4- ε)ln |V| for any ε>0, unless NP ⊆ DTIME(|V|^O(loglog|V|)).
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