Abstract
Abstract In this chapter we consider three easy cases of matchings in special graphs: trees, dense graphs and regular bipartite graphs. They are easy in the sense of possessing simple NC-algorithms for the matching problem. Usually efficient sequential algorithms for trees also have quite efficient parallel implementations. Regular bipartite and dense graphs are among the few known families of graphs which possess perfect matchings. A tree is an undirected connected graph without cycles. Assume we choose a vertex called a root and direct all edges top down. The root is the topmost vertex. The leaves are degree 1 vertices (except the root). A tree can have no perfect matching, even if the number of vertices is even. For example, K1,n has only one edge in the maximum cardinality matching. Our approach to construct an NC-algorithm for trees is to ca greedy sequential algorithm in parallel.
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