Three-fast-searchable graphs

Dereniowski, Dariusz; Diner, Öznur Yaşar; Dyer, Danny

doi:10.1016/j.dam.2013.03.004

Cited by 10 publications

(1 citation statement)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Dereniowski et al[6] independently proved the fast search problem is NP-complete by a ''weak search'' approach that is different from our method.…”

mentioning

confidence: 86%

Fast edge searching and fast searching on graphs

Yang

2011

Theoretical Computer Science

View full text Add to dashboard Cite

a b s t r a c tGiven a graph G = (V , E) in which a fugitive hides on vertices or along edges, graph searching problems are usually to find the minimum number of searchers required to capture the fugitive. In this paper, we consider the problem of finding the minimum number of steps to capture the fugitive. We introduce the fast edge searching problem in the edge search model, which is the problem of finding the minimum number of steps (called the fast edge-search time) to capture the fugitive. We establish relations between the fast edge searching and the fast searching that is the problem of finding the minimum number of searchers to capture the fugitive in the fast search model. While the family of graphs whose fast search number is at most k is not minor-closed for any positive integer k ≥ 2, we show that the family of graphs whose fast edge-search time is at most k is minor-closed. We establish relations between the fast (fast edge) searching and the node searching. These relations allow us to transform the problem of computing node search numbers to the problem of computing fast edge-search numbers or fast search numbers. Using these relations, we prove that the problem of deciding, given a graph G and an integer k, whether the fast (edge-)search number of G is less than or equal to k is NP-complete; and it remains NP-complete for Eulerian graphs. We also prove that the problem of determining whether the fast (edge-)search number of G is half of the number of odd vertices in G is NP-complete; and it remains NP-complete for planar graphs with maximum degree 4. We present a linear time approximation algorithm for the fast edge-search time that always delivers solutions of at most (1 + |V |−1 |E|+1 ) times the optimal value. This algorithm also gives us a tight upper bound on the fast search number of graphs. We also show a lower bound on the fast search number using the minimum degree and the number of odd vertices.

show abstract

“…Dereniowski et al[6] independently proved the fast search problem is NP-complete by a ''weak search'' approach that is different from our method.…”

mentioning

confidence: 86%

Fast edge searching and fast searching on graphs

Yang

2011

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

Network Decontamination

Nisse

2019

Distributed Computing by Mobile Entities

View full text Add to dashboard Cite

The Network Decontamination problem consists in coordinating a team of mobile agents in order to clean a contaminated network. The problem is actually equivalent to tracking and capturing an invisible and arbitrarily fast fugitive. This problem has natural applications in network security in computer science or in robotics for search or pursuitevasion missions. Many different objectives have been studied: the main one being the minimization of the number of mobile agents necessary to clean a contaminated network. Many environments (continuous or discrete) have also been considered. In this Chapter, we focus on networks modeled by graphs. In this context, the optimization problem that consists in minimizing the number of agents has a deep graph-theoretical interpretation. Network decontamination and, more precisely, graph searching models, provide nice algorithmic interpretations of fundamental concepts in the Graph Minors theory by Robertson and Seymour. For all these reasons, graph searching variants have been widely studied since their introduction by Breish (1967) and mathematical formalizations by and Petrov (1982). This chapter consists of an overview of algorithmic results on graph decontamination and graph searching.

show abstract