ALTERNATING CHAIN METHODS: A SURVEY

Abstract

This chapter describes alternating chain methods, wherein an alternating chain is a chain whose edges are alternately light and heavy. This concept was introduced in 1891 by Petersen to prove that, in some cubic graphs, any linear factor can be modified to use a given edge of the graph. To solve problems like when the set of vertices of a graph and the set of edges of it is different, the concept of alternating chain is offered. According to the theorems, the maximum matching problem can be solved by searching for all alternating chains from each unsaturated vertex. It is often advisable to return to a suitable fanning-out algorithm. For a better understanding of the procedure, consider, instead of G, a labeled graph H, obtained from G as follows. An alternating chain in G joining a to another unsaturated vertex is, in H, a path from a to z whose vertices are all marked differently and conversely. The problem is to find such a path which is called admissible. However, the maximum matching problem can be generalized. An alternating chain is not permitted to use the same edge more than once, but may visit the same vertex several times.