Abstract
A matching M in a graph G is an induced matching if the largest degree of the subgraph of G induced by M is equal to one. A dominating induced matching (DIM) of G is an induced matching that dominates every edge of G. It is well known that, if they exist, all dominating induced matchings of G are of the same size. The dominating induced matching number of G, denoted by dim ( G ) , is the size of any dominating induced matching of G. In this paper, we continue the study of dominating induced matchings. We prove that, if G has a DIM, then the induced matching number of G is equal to the independence number of its line graph L(G) and to the edge domination number of G. It is also shown that dim ( G ) ≤ 2 dim ( L ( G ) ) , provided that both G and L(G) have a DIM. We also present some bounds on dim ( G ) . In particular, for a tree T with a DIM we show that ⌈ n − l + 1 3 ⌉ ≤ dim ( T ) ≤ ⌊ n − 1 + l 3 ⌋ , where l is the number of leaves. Moreover, for a regular graph G we establish some Nordhaus-Gaddum type bounds.
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