Total domination in interval graphs
Information Processing Letters
On domination problems for permutation and other graphs
Theoretical Computer Science
Labeling algorithms for domination problems in sun-free chordal graphs
Discrete Applied Mathematics
Double total domination of graphs
Proceedings of an international symposium on Graphs and combinatorics
Some APX-completeness results for cubic graphs
Theoretical Computer Science
Introduction to algorithms
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Information Processing Letters
Hardness results and approximation algorithms of k-tuple domination in graphs
Information Processing Letters
Approximation hardness of dominating set problems in bounded degree graphs
Information and Computation
Networks
k-tuple total domination in graphs
Discrete Applied Mathematics
Total domination in block graphs
Operations Research Letters
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In this paper, we initiate the study of total liar's domination of a graph. A subset L⊆V of a graph G=(V,E) is called a total liar's dominating set of G if (i) for all v驴V, |N G (v)驴L|驴2 and (ii) for every pair u,v驴V of distinct vertices, |(N G (u)驴N G (v))驴L|驴3. The total liar's domination number of a graph G is the cardinality of a minimum total liar's dominating set of G and is denoted by 驴 TLR (G). The Minimum Total Liar's Domination Problem is to find a total liar's dominating set of minimum cardinality of the input graph G. Given a graph G and a positive integer k, the Total Liar's Domination Decision Problem is to check whether G has a total liar's dominating set of cardinality at most k. In this paper, we give a necessary and sufficient condition for the existence of a total liar's dominating set in a graph. We show that the Total Liar's Domination Decision Problem is NP-complete for general graphs and is NP-complete even for split graphs and hence for chordal graphs. We also propose a 2(lnΔ(G)+1)-approximation algorithm for the Minimum Total Liar's Domination Problem, where Δ(G) is the maximum degree of the input graph G. We show that Minimum Total Liar's Domination Problem cannot be approximated within a factor of $(\frac{1}{8}-\epsilon)\ln(|V|)$ for any ∈0, unless NP⊆DTIME(|V|loglog|V|). Finally, we show that Minimum Total Liar's Domination Problem is APX-complete for graphs with bounded degree 4.