Exploiting Restricted Linear Structure to Cope with the Hardness of Clique-Width

Clique-width is an important graph parameter whose computation is NP-hard. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any non-trivial graph class. Results so far indicate that proper interval graphs constitute the first interesting graph class on which we might have hope to compute clique-width, or at least its linear variant linear clique-width, in polynomial time. In TAMC 2009, a polynomialtime algorithm for computing linear clique-width on a subclass of proper interval graphs was given. In this paper, we present a polynomial-time algorithm for a larger subclass of proper interval graphs that approximates the clique-width within an additive factor 3. Previously known upper bounds on clique-width result in arbitrarily large difference from the actual clique-width when applied on this class. Our results contribute toward the goal of eventually obtaining a polynomial-time exact algorithm for clique-width on proper interval graphs.