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For a positive integer c, a graph G is said to be c-closed if every pair of nonadjacent vertices in G have at most c 1 neighbors in common. The closure of a graph G, denoted by cl(G), is the least positive integer c for which G is c-closed. The class of c-closed graphs was introduced by J. Fox, T. Roughgarden, C. Seshadhri, F. Wei, and N. Wein [Proceedings of the International Colloquium on Automata, Languages, and Programming (2018), 55; SIAM J. Comput., 49 (2020), pp. 448--464]. T. Koana, C. Komusiewicz, and F. Sommer [Proceedings of the European Symposium on Algorithms (2020), 65; SIAM J. Discrete Math., 36 (2022), pp. 2798--2821] started the study of using cl(G) as an additional structural parameter to design kernels for problems that are W-hard under standard parameterizations. In particular, they studied problems such as Independent Set, Induced Matching, Irredundant Set, and (Threshold) Dominating Set and showed that each of these problems admits a polynomial kernel when parameterized either by k + c or by k for each fixed value of c. Here, k is the solution size and c = cl(G). The work of Koana et al. left several questions open, one of which was whether the Perfect Code problem admits a fixed-parameter tractable (FPT) algorithm and a polynomial kernel on c-closed graphs. In this paper, among other results, we answer this question in the affirmative. Inspired by the FPT algorithm for Perfect Code, we further explore two more domination problems on the graphs of bounded closure. The other problems that we study are Connected Dominating Set and Partial Dominating Set. We show that Perfect Code and Connected Dominating Set are fixed-parameter tractable when parameterized by k + cl(G), whereas Partial Dominating Set, parameterized by k, is W[1]-hard even when cl(G) = 2. We also show that for each fixed c, Perfect Code admits a polynomial kernel on the class of c-closed graphs. And we observe that Connected Dominating Set has no polynomial kernel even on 2-closed graphs unless NP\subseteq co-NP/poly. [ABSTRACT FROM AUTHOR]

We prove the following theorem. Given a planar graph G and an integer k, it is possible in polynomial time to randomly sample a subset A of vertices of G with the following properties: A induces a subgraph of G of treewidth O(√k log k), and for every connected subgraph H of G on at most k vertices, the probability that A covers the whole vertex set of H is at least (2O(√k log² k)⋅nO(1))-1, where n is the number of vertices of G. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential-time parameterized algorithms for problems on planar graphs, usually with running time bound 2O(√k log² k)nO(1). The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph; examples of such problems include Directed k -Path, Weighted k -Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems could be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface. [ABSTRACT FROM AUTHOR]