We give an efficient algorithm for maintaining a minimum spanning forest of a plane graph subject to on-line modifications. The modifications supported include changes in the edge weights, and insertion and deletion of edges and vertices which are consistent with the given embedding. Our algorithm runs in O(log n) time per operation and O(n) space.

Introduction In many applications of graph algorithms, including communication networks, graphics, assembly planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of edges or vertices. In the last decade there has been a growing interest in such dynamically changing graphs, and a whole body of algorithms and data structures for dynamic graphs has been discovered. This chapter is intended as an overview of this field. In a typical dynamic graph problem one would like to answer queries on graphs that are undergoing a sequence of updates, for instance, insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update efficiently the solution of a problem after dynamic changes, rather than having to recompute it from scratch each time. Given their powerful versatility, it is not surprising that dynamic algorithms and dynamic data structures are often more difficult to design and analyze than their static c

The MAXIMUM PLANAR SUBGRAPH problem---given a graph G, find a largest planar subgraph of G---has applications in circuit layout, facility layout, and graph drawing. No previous polynomialtime approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1=3, which is achieved simply by producing a spanning tree of G. We present the first approximation algorithm for MAXIMUM PLANAR SUBGRAPH with higher performance ratio (4=9 instead of 1=3). We also apply our algorithm to find large outerplanar subgraphs. Last, we show that both MAXIMUM PLANAR SUBGRAPH and its complement, the problem of removing as few edges as possible to leave a planar subgraph, are Max SNP-Hard.

We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log 2 n) work per modification. No previous dynamic geometric MST algorithm was known.

We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a singlesource shortest path tree to changes in edge costs, and to analyze the sensitivity of a minimum cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heap-ordered queue data structure. Let G = (V; E) be a planar graph, either directed or undirected, with n vertices and m = O(n) edges. Each edge e 2 E has a real-valued cost cost(e). A minimum spanning tree of a connected, undirected planar graph G is a spanning tree of minimum total edge cost. If G is directed and r is a vertex from which all other vertices are reachable, then a shortest path tree from r is a spanning tree that contains a minimum-cost path from r to every...

We give improved solutions for the problem of generating the k smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes time O(m log #(m, n)+k 2 ); for planar graphs this bound can be improved to O(n + k 2 ). We also show that the k best spanning trees for a set of points in the plane can be computed in time O(min(k 2 n + n log n, k 2 + kn log(n/k))). The k best orthogonal spanning trees in the plane can be found in time O(n log n + kn log log(n/k)+k 2 ).

In the Minimum Label Spanning Tree problem

The Steiner tree problem in graphs requires to find a minimum size connected subgraph containing a given subset of nodes (given points). In this paper we consider this problem in three classes of graphs: where the maximum path distance is 2, where given points form a dominating set and where the given points form a vertex cover. As all these problems are MAX-SNP hard, the issue is what approximation can be obtained in polynomial time. In the first case we obtain an approximation ratio (of the achieved size over the minimal one) 5 4 + 3 80 = 1:2875, in the second case we achieve 4/3, and in the last case we achieve 8=7 \Gamma 1=160. 1 Introduction One of the strongest results in matroid theory is the polynomial time solution of the parity problem in linear matroids. Briefly stated, we have a collection of 2 \Theta n matrices, and our goal is to find a maximum size sub-collection such that they can be stacked into a single matrix of maximum rank. This problem was solved by Lov'asz [4]...

We provide the first nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NP-Hard problem of finding a heaviest planar subgraph in an edge-weighted graph G. This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH had performance ratio exceeding 1=3, which is obtained by any algorithm that produces a maximum weight spanning tree in G. Based on the Berman-Ramaiyer Steiner tree algorithm, the new algorithm has performance ratio at least 1/3 + 1/72. We also show that if G is complete and its edge weights satisfy the triangle inequality, then the performance ratio is at least 3/8. Furthermore, we derive the first nontrivial performance ratio (7/12 instead of 1/2) for the NP-Hard MAXIMUM WEIGHT OUTERPLANAR SUBGRAPH problem.

The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2=3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worst-case, while the bound for insertions is amortized. This is the first algorithm for this problem with sub-linear running time, and it affirmatively answers a question posed in [11]. The same data structure has further applications in maintaining the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n2=3) worst-case time per edge deletion, O(n2=3) amortized time per edge insertion, and O(n2=3) worst-case time to check whether two vertices are either biconnected or triconnected.