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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]

The Gallai graph Gal ( G ) of a graph G has the edges of G as its vertices, and two distinct vertices e and f of Gal ( G ) are adjacent in Gal ( G ) if the edges e and f of G are adjacent in G but do not span a triangle in G . In the present paper we characterize those graphs whose Gallai graphs are forests or trees, respectively. [ABSTRACT FROM AUTHOR]

Abstract: We prove that and for every triangle-free graph with maximum degree at most , where is the independence number and is the matching number of , respectively. These results are sharp for a graph on vertices. Furthermore we show for -free graphs, where is the chromatic number and is the clique number of , respectively. [Copyright &y& Elsevier]