Abstract. We obtain asymptotically tight algorithmic bounds for Max-Cut and Edge Dominating Set problems on graphs of bounded clique-width. We show that on an n-vertex graph of clique-width t both problems (1) cannot be solved in time f (t)n o(t) for any function f of t unless exponential time hypothesis fails, and (2) can be solved in time n O(t) . Key words. exponential time hypothesis, clique-width, max-cut, edge dominating set AMS subject classifications. 05C85, 68R10, 68Q17, 68Q25, 68W40 DOI. 10.1137/130910932 1. Introduction. Tree-width is one of the most fundamental parameters in graph algorithms. Graphs of bounded tree-width enjoy good algorithmic properties similar to trees, and this is why many problems which are hard on general graphs can be solved efficiently when the input is restricted to graphs of bounded tree-width. On the other hand, many hard problems also become tractable when restricted to graphs "similar to complete graphs." Courcelle and Olariu [6] introduced the notion of clique-width which captures nice algorithmic properties of both extremes. Since 2000, the research on algorithmic and structural aspects of clique-width is an active direction in graph algorithms, logic, and complexity. Corneil et al. [4] show that graphs of clique-width at most 3 can be recognized in polynomial time. By the meta-theorem of Courcelle, Makowsky, and Rotics [5], all problems expressible in M S 1 -logic (monadic second-order logic on graphs with quantification over subsets of vertices but not of edges) are fixed parameter tractable when parameterized by the clique-width of a graph and the expression size. For many other problems not expressible in this logic including problems like Max-Cut, Edge Dominating Set, Graph Coloring, or Hamiltonian Cycle, there is a significant amount of the literature devoted to algorithms for these problems and their generalizations on graphs of bounded clique-width
Fomin et al. REF proved that Maximum Cut is W[1]-hard but XP when parameterized by clique-width.
Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width
SIAM J. Comput.
SIAM J. Comput.
Mathematics Computer Science