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 Õ (1/ε13d6 + d/ε2 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 1/ε 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 Õ (1/ε13d6 + d/ε2. © 2010 Springer-Verlag.