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We continue the study initiated by Bonomo-Braberman and Gonzalez in 2020 on r-locally checkable problems. We propose a dynamic programming algorithm that takes as input a graph with an associated clique-width expression and solves a 1-locally checkable problem under certain restrictions. We show that it runs in polynomial time in graphs of bounded clique-width, when the number of colors of the locally checkable problem is fixed. Furthermore, we present a first extension of our framework to global properties by taking into account the sizes of the color classes, and consequently enlarge the set of problems solvable in polynomial time with our approach in graphs of bounded clique-width. As examples, we apply this setting to the k-Roman domination problem, as well as the k-community problem and some of its variants, thus providing the first linear and polynomial-time algorithms, respectively, in graphs of bounded clique-width for these problems.

A contact B 0 -VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-toone correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown in [15] that Recognition is NP-complete for contact B 0 -VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact B 0 -VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs.

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