Triangulations

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 decades 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 dynamicgraphproblemonewould like to answerqueries ongraphs that areundergoinga 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 counterparts. We can classify dynamic graph problems according to the types of updates allowed. A problemis said to be fully dynamic if the update operations include unrestricted insertions and deletions of edges. A problem is called partially dynamic if only one type of update, either insertions or deletions, is allowed. If only insertions are allowed, the problem is called incremental; if only deletions are allowed it is called decremental. In this chapter, we describe fully dynamic algorithms for graph problems. We present dynamicalgorithms for undirected graphs in Section 9.2. Dynamic algorithms for directed graphs are next described in Section 9.3. Finally, in Section 9.4 we describe some open problems.