Talk:Edge cover

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Greedy Algorithm

Latest comment: 16 years ago2 comments2 people in discussion

Can I get a citation for the greedy algorithm? I think it's not correct. Suppose a graph with four nodes V = {a,b,c,d} that form a line, i.e. E = {(a,b),(b,c),(c,d)}. Choosing (b,c) forms a maximal matching, but there is no way to greedily extend it to a minimal edge cover (that is {(a,b),(c,d)} in this graph). I googled around a bit but found nothing relevant. --134.96.156.86 (talk)

Never mind. I always get maximAL and maximUM matchings confused. --134.96.156.86 (talk) —Preceding undated comment added 11:40, 21 December 2009 (UTC). Reply

Right. Regarding citations, e.g. in you can find the theorem "the size of the minimum edge cover plus the size of the maximum number of independent edges equals the number of vertices of a graph"; note that "independent edges" = "matching". An easy way to prove this theorem is to consider the simple algorithm that greedily extends a maximum matching; once you have established the correctness of the algorithm, this theorem follows as a simple corollary (and vice versa). I think we could discuss this connection in more detail on this page. — Miym (talk) 17:54, 21 December 2009 (UTC)Reply