2009/09/23 by Ljiljana Branković, Henning Fernau, Brankovic, Ljiljana +4
Computer Science · #Advanced Graph Theory Research #Algorithms and Data Compression #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · pdf · doi:10.48550/arxiv.0909.4224
openalex publication_date 2009/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G) respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph parameters. It is a long-standing open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time less than the trivial Ω(2n) enumeration barrier. We solve these open problems by devising parameterized algorithms for the dual of the natural parameterizations of the problems with running times faster than O^*(4k). For example, we present an algorithm running in time O^*(3.069k) for determining whether IR(G) is at least n-k. Although the corresponding problem has been known to be in FPT by kernelization techniques, this paper offers the first parameterized algorithms with an exponential dependency on the parameter in the running time. Additionally, our work also appears to be the first example of a parameterized approach leading to a solution to a problem in exponential time algorithmics where the natural interpretation as an exact exponential-time algorithm fails.