2017/02/20 by Benjamin Bergougnoux, Bergougnoux, Benjamin, Mamadou Moustapha Kanté +3 · 4 citations
Computer Science · #Advanced Graph Theory Research #Algorithms and Data Compression #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.1702.06095
openalex publication_date 2017/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we prove that, given a clique-width k-expression of an n-vertex graph, Hamiltonian Cycle can be solved in time nO(k). This improves the naive algorithm that runs in time nO(k2) by Espelage et al. (WG 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time no(k) (SIAM. J. Computing 2014). We present a technique of representative sets using two-edge colored multigraphs on k vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on k vertices with at most n edges.