Well-indumatched trees and graphs of bounded girth

A graph G is called well-indumatched if all of its maximal induced match-ings have the same size. In this paper, we characterize all well-indumatch-ed trees. We provide a linear time algorithm to decide whether a tree is well-indumatched or not. Then, we characterize minimal well-indu-matched graphs of girth at least 9 and show subsequently that for an odd integer g & GE; 9 and g =6 11, there is no well-indumatched graph of girth g. On the other hand, there are infinitely many well-indumatched unicyclic graphs of girth k, where k & ISIN; {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.