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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 213084, 10 pages
http://dx.doi.org/10.5402/2011/213084
Research Article

An Efficient Algorithm to Solve the Conditional Covering Problem on Trapezoid Graphs

1Department of Mathematics, Narajole Raj College, Paschim Medinipur Narajole 721 211, India
2Department of Mathematics, National Institute of Technology, Durgapur 713209, India
3Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India

Received 16 July 2011; Accepted 29 August 2011

Academic Editors: Q. Gu, U. A. Rozikov, and R. Yeh

Copyright © 2011 Akul Rana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐺 = ( 𝑉 , 𝐸 ) be a simple connected undirected graph. Each vertex 𝑣 ∈ 𝑉 has a cost 𝑐 ( 𝑣 ) and provides a positive coverage radius 𝑅 ( 𝑣 ) . A distance 𝑑 u v is associated with each edge { 𝑒 , 𝑣 } ∈ 𝐸 , and 𝑑 ( 𝑒 , 𝑣 ) is the shortest distance between every pair of vertices 𝑒 , 𝑣 ∈ 𝑉 . A vertex 𝑣 can cover all vertices that lie within the distance 𝑅 ( 𝑣 ) , except the vertex itself. The conditional covering problem is to minimize the sum of the costs required to cover all the vertices in 𝐺 . This problem is NP-complete for general graphs, even it remains NP-complete for chordal graphs. In this paper, an 𝑂 ( 𝑛 2 ) time algorithm to solve a special case of the problem in a trapezoid graph is proposed, where 𝑛 is the number of vertices of the graph. In this special case, 𝑑 u v = 1 for every edge { 𝑒 , 𝑣 } ∈ 𝐸 , 𝑐 ( 𝑣 ) = 𝑐 for every 𝑣 ∈ 𝑉 ( 𝐺 ) , and 𝑅 ( 𝑣 ) = 𝑅 , an integer >1, for every 𝑣 ∈ 𝑉 ( 𝐺 ) . A new data structure on trapezoid graphs is used to solve the problem.