Core Concepts
The computational complexity of determining the interval number, hull number, rank, and convexity number in the cycle convexity of graphs is analyzed, showing that these problems are NP-complete and W[1]-hard.
Abstract
The content explores the computational complexity of various parameters in the cycle convexity of graphs, which is a recently defined graph convexity concept.
Key highlights:
The interval number in cycle convexity can be computed in linear time for graphs with two universal vertices, but is NP-complete and W[2]-hard for bipartite graphs.
The hull number in cycle convexity is NP-complete even for simple planar graphs.
The rank in cycle convexity is NP-complete and W[1]-hard, as it is equivalent to the problem of finding the maximum induced forest.
The convexity number in cycle convexity is NP-complete and W[1]-hard.
The percolation time, which is the largest integer k such that there exists a hull set S with Ik-1(S) ≠ V, is NP-complete for any fixed k ≥ 9, but can be computed in cubic time for k = 2.
The article provides detailed proofs and analysis for establishing these computational complexity results, demonstrating the inherent difficulty of working with cycle convexity parameters.