Well-indumatched Trees and Graphs of Bounded Girth
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A graph G is called well-indumatched if all of its maximal induced matchings have the same size. In this paper we characterize all well-indumatched trees. We provide a linear time algorithm to decide if a tree is well-indumatched or not. Then, we characterize minimal well-indumatched graphs of girth at least 9 and show subsequently that there is no well-indumatched graph of odd girth g greater than or equal to 9 and different from 11. On the other hand, there are infinitely many well-indumatched unicyclic graphs of girth k, where k is in 3, 5, 7 or k is an even integer greater than 2. We also show that, although the recognition of well-indumatched graphs is known to be co-NP-complete in general, one can recognize in polynomial time well-indumatched graphs where the size of maximal induced matchings is fixed.
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