A unit disk graph is the intersection graph of a set of unit diameter disks in the plane. In this paper we consider liar's domination problem on unit disk graphs, a variant of dominating set problem. We call this problem as Euclidean liar's domination problem . In the Euclidean liar's domination problem, a set @math of @math points (disk centers) are given in the Euclidean plane. For @math , @math is a subset of @math such that for any @math , the Euclidean distance between @math and @math is less than or equal to 1, i.e., the corresponding unit diameter disks intersect. The objective of the Euclidean liar's domination problem is to find a subset @math of minimum size having the following properties : (i) @math for @math , and (ii) @math for @math . This article aims to prove the Euclidean liar's domination problem is NP-complete.
The liar's domination problem is introduced by Slater in 2009 and showed that the problem is NP-hard for general graphs @cite_2 . Later, Roden and Slater showed that the problem is NP-hard even for bipartite graphs @cite_6 . Panda and Paul @cite_3 proved that the problem is NP-hard for split graphs and chordal graphs. The authors also proposed a linear time algorithm for computing a minimum cardinality liar's dominating set in a tree.
@cite_3 @cite_6 @cite_2
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A set L@?V(G) of a graph G=(V,E) is a liar's dominating set if (1) for all v@?V(G), |N"G[v]@?L|>=2 and (2) for every pair u,v@?V(G) of distinct vertices, |(N"G[u]@?N"G[v])@?L|>=3. A graph G=(V,E) admits a liar's dominating set if each of its connected component has at least three vertices. Given a graph G=(V,E) and an integer K, the liar's domination decision problem (LR-DOMDP) is to decide whether G has a liar's dominating set of cardinality at most K. Slater [P.J. Slater, Liar's Domination, Networks, 54(2) (2009) 70-74] proved that the LR-DOMDP is NP-complete for general graphs. Subsequently, Roden and Slater [M.L. Roden and P.J. Slater, Liar's domination in graphs, Discrete Math., 309(19) (2009) 5884-5890] showed a more general family of problems to each be NP-complete for bipartite graphs. Besides this result, no other algorithmic result for the liar's dominating set problem is available in the literature. In this paper, we first strengthen the complexity result of the LR-DOMDP by showing that this problem remains NP-complete for split graphs and hence for chordal graphs. Finally, we propose a linear time algorithm for computing a minimum cardinality liar's dominating set in a tree. Let G=(V,E) be a graph without isolated vertices and having at least 3 vertices. A set L@?V(G) is a liar@?s dominating set if (1) |N"G[v]@?L|>=2 for all v@?V(G), and (2) |(N"G[u]@?N"G[v])@?L|>=3 for every pair u,v@?V(G) of distinct vertices in G, where N"G[x]= y@?V|xy@?E @? x is the closed neighborhood of x in G. Given a graph G and a positive integer k, the liar@?s domination problem is to check whether G has a liar@?s dominating set of size at most k. The liar@?s domination problem is known to be NP-complete for general graphs. In this paper, we propose a linear time algorithm for computing a minimum cardinality liar@?s dominating set in a proper interval graph. We also strengthen the NP-completeness result of liar@?s domination problem for general graphs by proving that the problem remains NP-complete even for undirected path graphs which is a super class of proper interval graphs. A subset L ? V of a graph G = ( V , E ) is called a liar's dominating set of G if (i) | N G u ] ? L | ? 2 for every vertex u ? V , and (ii) | ( N G u ] ? N G v ] ) ? L | ? 3 for every pair of distinct vertices u , v ? V . The Min Liar Dom Set problem is to find a liar's dominating set of minimum cardinality of a given graph G and the Decide Liar Dom Set problem is the decision version of the Min Liar Dom Set problem. The Decide Liar Dom Set problem is known to be NP-complete for general graphs. In this paper, we first present approximation algorithms and hardness of approximation results of the Min Liar Dom Set problem in general graphs, bounded degree graphs, and p-claw free graphs. We then show that the Decide Liar Dom Set problem is NP-complete for doubly chordal graphs and propose a linear time algorithm for computing a minimum liar's dominating set in block graphs. Introduction (S.T. Hedetniemi, R.C. Laskar). Theoretical. Chessboard Domination Problems (E.J. Cockayne). On the Queen Domination Problem (C.M. Grinstead, B. Hahne, D. Van Stone). Recent Problems and Results About Kernels in Directed Graphs (C. Berge, P. Duchet). Critical Concepts in Domination (D.P. Summer). The Bondage Number of a Graph (J.F. ). Chordal Graphs and Upper Irredundance, Upper Domination and Independence (M.S. Jacobson and K. Peters). Regular Totally Domatically Full Graphs (B. Zelinka). Domatically Critical and Domatically Full Graphs (D. Rall). On Generalised Minimal Domination Parameters for Paths (B. Bollobas, E.J. Cockayne, C.M. Mynhardt). New Models. Dominating Cliques in Graphs (M.B. Cozzens, L.L. Kelleher). Covering all Cliques of Graph (Z. Tuza). Factor Domination in Graphs (R.C. Brigham, R. Dutton). The Least Point Covering and Domination Numbers of a Graph (E. Sampathkumar). Algorithmic. Dominating Sets in Perfect Graphs (D.G. Corneil, L.K. Stewart). Unit Disk Graphs (B.N. Clark, C.J. Colbourn, D.S. Johnson). Permutation Graphs: Connected Domination and Steiner Trees (C.J. Colbourn, L.K. Stewart). The Discipline Number of a Graph (V. Chvatal, W.J. Cook). Best Location of Service Centers in a Tree-Like Network under Budget Constraints (J. McHugh, Y. Perl). Dominating Cycles in Halin Graphs (M. Skowronska, M.M. Syslo). Finding Dominating Cliques Efficiently, in Strongly Chordal Graphs and Undirected Path Graphs (D. Kratsch). On Minimum Dominating Sets with Minimum Intersection (D.L. Grinstead, P.J. Slater). Bibliography. Bibliography on Domination in Graphs and Some Basic Definitions of Domination Parameters (S.T. Hedetniemi, R.C. Laskar).