Decomposing a triangle-free planar graph into a forest and a subcubic forest

We strengthen a result of Dross, Montassier and Pinlou (2017) that the vertex set of every triangle-free planar graph can be decomposed into a set that induces a forest and a set that induces a forest with maximum degree at most $5$, showing that $5$ can be replaced by $3$.Comment: 8 pages. This version corrects an error in the previous version (where we made a false claim at the end of the proof of Theorem 2), without changing the overall structure of the proo

Computing the Girth of a Planar Graph in Linear Time

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin