Subhabrata Paul

A linear time algorithm for liar's domination problem in proper interval graphs

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) |NG[v]@?L|>=2 for all v@?V(G), and (2) |(NG[u]@?NG[v])@?L|>=3 for every pair u,v@?V(G) of distinct vertices in G, where NG[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.

Liar's domination in graphs: Complexity and algorithm

A set L@?V(G) of a graph G=(V,E) is a liars dominating set if (1) for all v@?V(G), |NG[v]@?L|>=2 and (2) for every pair u,v@?V(G) of distinct vertices, |(NG[u]@?NG[v])@?L|>=3. A graph G=(V,E) admits a liars dominating set if each of its connected component has at least three vertices. Given a graph G=(V,E) and an integer K, the liars domination decision problem (LR-DOMDP) is to decide whether G has a liars dominating set of cardinality at most K. Slater [P.J. Slater, Liars 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, Liars 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 liars 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 liars dominating set in a tree.

Algorithmic Aspects of Disjunctive Domination in Graphs

For a graph (G=(V,E)), a set (Dsubseteq V) is called a disjunctive dominating set of G if for every vertex (vin Vsetminus D), v is either adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The cardinality of a minimum disjunctive dominating set of G is called the disjunctive domination number of graph G, and is denoted by (gamma _{2}^{d}(G)). The Minimum Disjunctive Domination Problem (MDDP) is to find a disjunctive dominating set of cardinality (gamma _{2}^{d}(G)). Given a positive integer k and a graph G, the Disjunctive Domination Decision Problem (DDDP) is to decide whether G has a disjunctive dominating set of cardinality at most k. In this article, we first propose a polynomial time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a ((ln (Delta ^{2}+Delta +2)+1))-approximation algorithm for MDDP, where (Delta ) is the maximum degree of input graph (G=(V,E)) and prove that MDDP can not be approximated within ((1-epsilon ) ln (|V|)) for any (epsilon >0) unless NP (subseteq ) DTIME((|V|^{O(log log |V|)})). Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree 3.

Hardness results and approximation algorithm for total liar's domination in graphs

In this paper, we initiate the study of total liar’s domination of a graph. Axa0subset L⊆V of a graph G=(V,E) is called a total liar’s dominating set of G if (i) for all v∈V, |NG(v)∩L|≥2 and (ii) for every pair u,v∈V of distinct vertices, |(NG(u)∪NG(v))∩L|≥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 γTLR(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

Uniformity of Point Samples in Metric Spaces Using Gap Ratio

(frac{1}{8}-epsilon)ln(|V|)

( 1 , j ) -set problem in graphs

for any ϵ>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.

Hardness results, approximation and exact algorithms for liar's domination problem in graphs

Teramoto et al. [22] defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from (mathcal S), a bounded subset of (mathbb {R}^2). We attempt to generalize the definition of this measure over all metric spaces. We solve optimization related questions about selecting uniform point samples from metric spaces; the uniformity is measured using gap ratio. We give lower bounds for specific metric spaces, prove hardness and approximation hardness results. We also give a general approximation algorithm framework giving different approximation ratios for different metric spaces and give a (left( 1+epsilon right) )-approximation algorithm for a set of points in a Euclidean space.

Linear kernels for k-tuple and liar's domination in bounded genus graphs

A subset D ź V of a graph G = ( V , E ) is a ( 1 , j ) -set (Chellali etźal., 2013) if every vertex v ź V ź D is adjacent to at least 1 but not more than j vertices in D . The cardinality of a minimum ( 1 , j ) -set of G , denoted as γ ( 1 , j ) ( G ) , is called the ( 1 , j ) -domination number of G . In this paper, using probabilistic methods, we obtain an upper bound on γ ( 1 , j ) ( G ) for j ź O ( log Δ ) , where Δ is the maximum degree of the graph. The proof of this upper bound yields a randomized linear time algorithm. We show that the associated decision problem is NP-complete for choral graphs but, answering a question of Chellali etźal., provide a linear-time algorithm for trees for a fixed j . Apart from this, we design a polynomial time algorithm for finding γ ( 1 , j ) ( G ) for a fixed j in a split graph, and show that ( 1 , j ) -set problem is fixed parameter tractable in bounded genus graphs and bounded treewidth graphs.

Parameterized complexity of k-tuple and liar's domination.

A subset L ? V of a graph G = ( V , E ) is called a liars 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 liars 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 liars dominating set in block graphs.

Grid obstacle representation of graphs.

Abstract A set D ⊆ V is called a k -tuple dominating set of a graph G = ( V , E ) if | N G [ v ] ∩ D | ≥ k for all v ∈ V , where N G [ v ] denotes the closed neighborhood of v . A set D ⊆ V is called a liar’s dominating set of a graph G = ( V , E ) if (i) | N G [ v ] ∩ D | ≥ 2 for all v ∈ V and (ii) for every pair of distinct vertices u , v ∈ V , | ( N G [ u ] ∪ N G [ v ] ) ∩ D | ≥ 3 . Given a graph G , the decision versions of k - Tuple Domination Problem and the Liar’s Domination Problem are to check whether there exist a k -tuple dominating set and a liar’s dominating set of G of a given cardinality, respectively. These two problems are known to be NP -complete (Liao and Chang, 2003; Slater, 2009). In this paper, we study the parameterized complexity of these problems. We show that the k - Tuple Domination Problem and the Liar’s Domination Problem are W[2] -hard for general graphs. It can be verified that both the problems have a finite integer index and satisfy certain coverability property. Hence they admit linear kernel as per the meta-theorem in Bodlaender (2009), but the meta-theorem says nothing about the constant. In this paper, we present a direct proof of the existence of linear kernel with small constants for both the problems.