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For a graph G, let vs(G) be the size of a largest induced matching of G. We prove that vs(G) ≥ n(G)/(⌈Δ/2⌇+1)(⌈Δ/2⌇+1) for every graph of sufficiently large maximum degree Δ and without isolated vertices. This bound is sharp. Moreover, there is a polynomial-time algorithm which computes induced matchings of the size stated above. [ABSTRACT FROM AUTHOR]

For a graph G, let νs(G) be the strong matching number of G. We prove the sharp bound νs(G) ≥ n(G)/9 for every graph G of maximum degree at most 4 and without isolated vertices that does not contain a certain blown-up 5-cycle as a component. This result implies a strengthening of a consequence, namely, νs(G) ≥ m(G)/18 for such graphs and νs(G) ≥ m(G)/20 for a graph of maximum degree 4, of the well-known conjecture of Erdős and Nešetřil, which says that the strong chromatic index χ's(G) of a graph G is at most 5/4Δ(G)², since νs(G) ≥ m(G)/χ's(G) and n(G) ≥ 2m(G)/Δ(G). This bound is tight and the proof implies a polynomial time algorithm to find an induced matching of this size. [ABSTRACT FROM AUTHOR]

We characterize the graphs for which the independence number equals the packing number. As a consequence we obtain simple structural descriptions of the graphs for which (i) the distance- k -packing number equals the distance- 2 k -packing number, and (ii) the distance- k -matching number equals the distance- 2 k -matching number. This last result considerably simplifies and extends previous results of Cameron and Walker (2005). For positive integers k 1 and k 2 with k 1 < k 2 and ⌈ ( 3 k 2 + 1 ) / 2 ⌉ ≤ 2 k 1 + 1 , we prove that it is NP-hard to determine for a given graph whether its distance- k 1 -packing number equals its distance- k 2 -packing number. [ABSTRACT FROM AUTHOR]

Kobler and Rotics in 2003, and Cameron and Walker in 2005, gave a complete structural description of the graphs G where the matching number ν ( G ) equals the induced matching number ν 2 ( G ) . We study their result and use it to analyse graphs G with ν ( G ) − ν 2 ( G ) ≤ k . We show that the recognition of these graphs can be done in polynomial time for fixed k , and is fixed parameter tractable when parameterized by k for graphs of bounded maximum degree. [ABSTRACT FROM AUTHOR]

Extending results of Kobler and Rotics (2003), Cameron and Walker (2005) gave a complete structural description of the graphs G where the matching number ν ( G ) equals the induced matching number ν 2 ( G ) . We present a short proof of their result and use it to study graphs G with ν ( G ) − ν 2 ( G ) ≤ k . We show that the recognition of these graphs can be done in polynomial time for fixed k , and is fixed parameter tractable when parameterized by k for graphs of bounded maximum degree. [ABSTRACT FROM AUTHOR]

We prove that a cubic graph with m edges has an induced matching with at least M/9 edges. Our result generalizes a result for planar graphs due to Kang, Mnich, and Müller (SIAM J. Discrete Math., 26 (2012), pp. 1383-1411) and solves a conjecture of Henning and Rautenbach. [ABSTRACT FROM AUTHOR]