Tight Algorithm for Connected Odd Cycle Transversal Parameterized by Clique-width

Recently, Bojikian and Kratsch [2023] have presented a novel approach to tackle connectivity problems parameterized by clique-width (cw), based on counting small representations of partial solutions (modulo two). Using this technique, they were able to get a tight bound for the Steiner Tree problem, answering an open question posed by Hegerfeld and Kratsch [ESA, 2023]. We use the same technique to solve the Connected Odd Cycle Transversal problem in time 𝒪^*(12^cw). We define a new representation of partial solutions by separating the connectivity requirement from the 2-colorability requirement of this problem. Moreover, we prove that our result is tight by providing SETH-based lower bound excluding algorithms with running time 𝒪^*((12-ϵ)^lcw) even when parameterized by linear clique-width. This answers the second question posed by Hegerfeld and Kratsch in the same paper.