Why certain subgraph computations requite only linear time

Abstract

A general problem in computational graph theory is that of finding an optimal subgraph H of a given weighted graph G. The matching problem (which is easy) and the traveling salesman problem (which is not) are well known examples of this general problem. In the literature one can also find a variety of ad hoc algorithms for solving certain special cases in linear time. We present a general methodology for constructing linear time algorithms in the case that the graph G is defined by certain rules of composition (as are trees, series parallel graphs, and outerplanar graphs) and the desired subgraph H satisfies a "regular" property (such as independence or matching). This methodology is applied to obtain a linear time algorithm for computing the irredundance number of a tree, a problem for which no polynomial time algorithm was previously known.