33 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
A Planarity Test via Construction Sequences
Optimal linear-time algorithms for testing the planarity of a graph are
well-known for over 35 years. However, these algorithms are quite involved and
recent publications still try to give simpler linear-time tests. We give a
simple reduction from planarity testing to the problem of computing a certain
construction of a 3-connected graph. The approach is different from previous
planarity tests; as key concept, we maintain a planar embedding that is
3-connected at each point in time. The algorithm runs in linear time and
computes a planar embedding if the input graph is planar and a
Kuratowski-subdivision otherwise
Improved Algorithms for Dynamic Shortest Paths
We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time. A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting tradeoff between preprocessing, query, and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to n-vertex digraphs of genus O(n1−ε) for any ε>0