Embeddings of graphs with no short noncontractible cycles
Journal of Combinatorial Theory Series B
The problem of compatible representatives
SIAM Journal on Discrete Mathematics
Surface Approximation and Geometric Partitions
SIAM Journal on Computing
Barrier coverage with wireless sensors
Proceedings of the 11th annual international conference on Mobile computing and networking
Minimum-weight triangulation is NP-hard
Journal of the ACM (JACM)
Approximating Barrier Resilience in Wireless Sensor Networks
Algorithmic Aspects of Wireless Sensor Networks
Finding shortest non-trivial cycles in directed graphs on surfaces
Proceedings of the twenty-sixth annual symposium on Computational geometry
On isolating points using disks
ESA'11 Proceedings of the 19th European conference on Algorithms
SIAM Journal on Computing
On barrier resilience of sensor networks
ALGOSENSORS'11 Proceedings of the 7th international conference on Algorithms for Sensor Systems, Wireless Ad Hoc Networks and Autonomous Mobile Entities
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We study the following separation problem: Given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining, in this graph, an appropriate family of closed walks that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here reveals the connection to topology. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles.