# An optimal XP algorithm for Hamiltonian cycle on graphs of bounded clique-width

## TL;DR

This paper introduces an optimal algorithm for solving the Hamiltonian Cycle problem on graphs with bounded clique-width, improving efficiency and matching theoretical lower bounds.

## Contribution

It presents a new algorithm with a novel technique using representative sets and two-edge colored multigraphs, achieving optimal runtime for graphs of bounded clique-width.

## Key findings

- Algorithm runs in time n^{O(k)} for graphs with clique-width k.
- Improves upon previous algorithms with time n^{O(k^2)}.
- Matches the lower bound under the Exponential Time Hypothesis.

## Abstract

In this paper, we prove that, given a clique-width $k$-expression of an $n$-vertex graph, \textsc{Hamiltonian Cycle} can be solved in time $n^{\mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{\mathcal{O}(k^2)}$ 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 $n^{o(k)}$ (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.