Subexponential Parameterized Algorithms for Bounded-Degree Connected Subgraph Problems on Planar Graphs
Abstract
We present subexponential parameterized algorithms on planar graphs for a family of problems that consist in, given a graph $G$, finding a connected subgraph $H$ with bounded maximum degree, while maximising the number of edges (or vertices) of $H$. These problems are natural generalisations of the \\textscLongest Path problem. Our approach uses bidimensionality theory to obtain combinatorial bounds, combined with dynamic programming techniques over a branch decomposition of the input graph. These techniques need to be able to keep track of the connected components of the partial solutions over the branch decomposition, and can be seen as an \\emphalgorithmic tensor that can be applied to a wide family of problems that deal with finding connected subgraphs under certain constraints.