7 research outputs found
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
Topologically Trivial Closed Walks in Directed Surface Graphs
Let be a directed graph with vertices and edges, embedded on a
surface , possibly with boundary, with first Betti number . We
consider the complexity of finding closed directed walks in that are either
contractible (trivial in homotopy) or bounding (trivial in integer homology) in
. Specifically, we describe algorithms to determine whether contains a
simple contractible cycle in time, or a contractible closed walk in
time, or a bounding closed walk in time. Our
algorithms rely on subtle relationships between strong connectivity in and
in the dual graph ; our contractible-closed-walk algorithm also relies on
a seminal topological result of Hass and Scott. We also prove that detecting
simple bounding cycles is NP-hard.
We also describe three polynomial-time algorithms to compute shortest
contractible closed walks, depending on whether the fundamental group of the
surface is free, abelian, or hyperbolic. A key step in our algorithm for
hyperbolic surfaces is the construction of a context-free grammar with
non-terminals that generates all contractible closed walks of
length at most L, and only contractible closed walks, in a system of quads of
genus . Finally, we show that computing shortest simple contractible
cycles, shortest simple bounding cycles, and shortest bounding closed walks are
all NP-hard.Comment: 30 pages, 18 figures; fixed several minor bugs and added one figure.
An extended abstraction of this paper will appear at SOCG 201