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Tutte sets in graphs I: Maximal tutte sets and D-graphs
Article first published online: 29 MAR 2007
DOI: 10.1002/jgt.20243
Copyright © 2007 Wiley Periodicals, Inc.
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How to Cite
Bauer, D., Broersma, H. J., Morgana, A. and Schmeichel, E. (2007), Tutte sets in graphs I: Maximal tutte sets and D-graphs. Journal of Graph Theory, 55: 343–358. doi: 10.1002/jgt.20243
Publication History
- Issue published online: 12 JUN 2007
- Article first published online: 29 MAR 2007
- Manuscript Revised: 14 NOV 2006
- Manuscript Received: 12 MAY 2005
- Abstract
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- Cited By
Keywords:
- perfect matching;
- Tutte set;
- extreme set;
- deficiency;
- independent set;
- D-graph
Abstract
A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency of G. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. In this article, we study the structural aspects of maximal Tutte sets in a graph G. Towards this end, we introduce a related graph D(G). We first show that the maximal Tutte sets in G are precisely the maximal independent sets in its D-graph D(G), and then continue with the study of D-graphs in their own right, and of iterated D-graphs. We show that G is isomorphic to a spanning subgraph of D(G), and characterize the graphs for which G≅D(G) and for which D(G)≅D2(G). Surprisingly, it turns out that for every graph G with a perfect matching, D3(G)≅D2(G). Finally, we characterize bipartite D-graphs and comment on the problem of characterizing D-graphs in general. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 343–358, 2007