Testing Outerplanarity of Bounded Degree Graphs

Abstract

We present an efficient algorithm for testing outerplanarity of graphs in the bounded degree model. In this model, given a graph G with n vertices and degree bound d, we should distinguish with high probability the case that G is outerplanar from the case that modifying at least an ε-fraction of the edge set of G is necessary to make G outerplanar.

Our algorithm runs in ${\tilde{O}\left(\frac{1}{\epsilon^{13}d^6}+\frac{d}{\epsilon^2}\right)}$ time, which is independent of the size of graphs. This is the first algorithm for a non-trivial minor-closed property whose time complexity is polynomial in $\frac{1}{\epsilon}$ and d. To achieve the time complexity, we exploit the tree-like structure inherent to an outerplanar graph using the microtree/macrotree decomposition of a tree.

As a corollary, we also show an algorithm that tests whether a given graph is a cactus with time complexity ${\tilde{O}\left(\frac{1}{\epsilon^{13}d^6}+\frac{d}{\epsilon^2}\right)}$.