Vertex-independent sets and vertex coverings as also edge-independent sets and edge coverings of graphs occur very naturally in many practical situations and hence have several potential applications. In this chapter, we study the properties of these sets. In addition, we discuss matchings in graphs and, in particular, in bipartite graphs. Matchings in bipartite graphs have varied applications in operations research. We also present two celebrated theorems of graph theory, namely, Tutte's 1-factor theorem and Hall's matching theorem. All graphs considered in this chapter are loopless.
Vertex-Independent Sets and Vertex CoveringsDefinition 5.2.1. A subset S of the vertex set V of a graph G is called independent if no two vertices of S are adjacent in G. S Â V is a maximum independent set of G if G has no independent set S 0 with jS 0 j > jS j. A maximal independent set of G is an independent set that is not a proper subset of another independent set of G.For example, in the graph of Fig. 5.1, fu; v; wg is a maximum independent set and fx; yg is a maximal independent set that is not maximum.
Definition 5.2.2.A subset K of V is called a covering of G if every edge of G is incident with at least one vertex of K. A covering K is minimum if there is no covering K 0 of G such that jK 0 j < jKjI it is minimal if there is no covering K 1 of G such that K 1 is a proper subset of K.In the graph W 5 of Fig. 5.2, fv 1 ; v 2 ; v 3 ; v 4 ; v 5 g is a covering of W 5 and fv 1 ; v 3 ; v 4 ; v 6 g is a minimal covering. Also, the set fx; yg is a minimum covering of the graph of Fig. 5.1.
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