ISRN Discrete Mathematics

Volume 2011, Article ID 213084, 10 pages

http://dx.doi.org/10.5402/2011/213084

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

^{1}Department of Mathematics, Narajole Raj College, Paschim Medinipur Narajole 721 211, India^{2}Department of Mathematics, National Institute of Technology, Durgapur 713209, India^{3}Department 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 be a simple connected undirected graph. Each vertex has a cost and provides a positive coverage radius . A distance is associated with each edge and is the shortest distance between every pair of vertices . A vertex can cover all vertices that lie within the distance , except the vertex itself. The conditional covering problem is to minimize the sum of the costs required to cover all the vertices in . This problem is NP-complete for general graphs, even it remains NP-complete for chordal graphs. In this paper, an time algorithm to solve a special case of the problem in a trapezoid graph is proposed, where is the number of vertices of the graph. In this special case, for every edge , for every and , an integer >1, for every . A new data structure on trapezoid graphs is used to solve the problem.