Computing the Girth of a Planar Graph in O(n logn) Time

Abstract

Abstract. We give an O(n log n) algorithm for computing the girth (shortest cycle) of an undirected n-vertex planar graph. Our solution extends to any graph of bounded genus. This improves upon the best previously known algorithms for this problem.Key words. Girth, shortest cycle, planar graph, graphs of bounded genus AMS subject classifications. 05C38, 68R101. Introduction. The girth of a graph is the length of its shortest cycle, or infinity if the graph does not contain any cycles. In addition to being a basic combinatorial characteristic of graphs, the girth has tight connections to many other graph properties. The connection between the girth of a graph and its chromatic number was studied by Erdős [13], Lovasz [19], Bollobás [4], and Cook [6]. Other important graph properties related to the girth include the minimum or average degree of the vertices, the diameter, the connectivity, the maximum genus, and the existence of certain type of minors (see Diestel's book [8] for a review of results).The problem of computing the girth of a graph is among the most natural and easily stated algorithmic graph problems. Itai and Rodeh [17] were the first to suggest an efficient algorithm to compute the girth. They presented an O(nm)-time algorithm for a graph of n vertices and m edges, and an O(n 2 )-time algorithm if an additive error of one is allowed. Monien [20] showed that finding the shortest cycle of even length is easier and can be done in O(n 2 α(n)) time, where α(n) is the inverse Ackermann function. For the case of planar graphs, Eppstein [11] proved that the girth can be found in O(n) time provided it is bounded by some constant. His result extended that of Itai and Rodeh [17] and of Papadimitriou and Yannakakis [21] who proved this for girth bounded by 3. For the general case, when the girth is not bounded by a constant, Djidjev [9] presented and algorithm that computes the girth in O(n 5/4 log n) time. Djidjev's solution uses dynamic data structure for shortest paths [10], as well as a clever use of hammock decompositions [14]. Djidjev's algorithm is the fastest algorithm that solves this problem directly. However, there is another, indirect approach to solve the girth problem in planar graphs. It is a known fact that cuts in an embedded planar graph correspond to cycles in the dual plane graph. Furthermore, minimum cuts correspond to shortest cycles in the dual plane graph. Chalermsook et al. [5] gave an O(n log 2 n) time algorithm for the minimum-cut problem in planar graphs. This algorithm can be used to solve the girth problem in planar graphs with positive edge weights in the same time, by reducing it (in linear time) to the min-cut problem in planar graphs. We note that this reduction introduces (not necessarily constant) weights in the dual graph even if the original graph was unweighted.

Min-Cuts and Shortest Cycles in Planar Graphs in O(n loglogn) Time

“…Here we show an even further improvement by showing an O(n log log n) time algorithm for both minimum cut and shortest cycle problem in weighted planar graphs. This not only improves over the result in [9] but also over the result of Weimann an Yuster [15] for unweighted graphs. The minimum cut problem is related to minimum st-cut problem, where we need to find minimum cut that separates s from t. Up until the paper of Italiano et al [9] the fastest known algorithm worked in O(n log n) time [6].…”

“…This actually implies a linear time algorithm for minimum cuts in planar unweighted graphs, as the sizes of the minimum cuts are at most 5. On the other hand, the fastest algorithm for finding the shortest cycle in unweighted graph (also called the girth of the graph), was given by Weimann and Yuster [15]. This very recent algorithm works in O(n log n) time.…”

Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Noncrossing Shortest Paths

“…If G is undirected and unweighted and is 2-cell embedded on an orientable surface of genus g = O(n α ) with 0 < α < 1, Djidjev [12] solved the problem in O(g 3/4 n 5/4 log n) time. On undirected unweighted O(1)-genus G, Weimann and Yuster [67] solved the problem in O(n log n) time. For directed planar G, even if G is unweighted, our Theorem 1.1 remains the best known algorithm.…”

“…Djidjev [12] claimed that his technique for unweighted undirected planar graphs solves the shortest-cycle problem on unweighted directed planar G in O(n 3/2 ) time and left open the problem of finding a shortest cycle in unweighted directed planar G in o(n 3/2 ) time. Weimann and Yuster [67] gave an O(n 3/2 )-time algorithm for the shortest-cycle problem, which should be adjustable to solve the minimum-cut problem also in O(n 3/2 ) time (via similar techniques to our proof for Lemma 4.2 in §4 to handle degeneracy in shortest cycles). Wulff-Nilsen [68] reduced the time for the shortest-cycle problem on G to O(n log 3 n), but it is unclear how to adjust his algorithm to solve the minimum-cut problem without increasing the required time by too much.…”

Computing the Girth of a Planar Graph in Linear Time

“…Otherwise, by girth(G ′ ) = O(log 2 m), as proved by Weimann and Yuster (see Lemma 2.4), and the fact G ′ has positive integral weights, we can further transform G ′ to a Θ(m)-node O(log 2 m)-outerplane graph G with O(1) degree, O(log 2 m) density, and O(log 2 m) maximum weight such that girth(G) = girth(G ′ ). The way we reduce the "outerplane radius" (see §2.2) is similar to those of Djidjev [17] and Weimann and Yuster [49]. In order not to increase the outerplane radius, our degree-reduction operation (see §2.2) is different from that of Djidjev [17].…”