A graph G is called H‐saturated if it does not contain any copy of H, but for any edge e in the complement of G, the graph G+e contains some H. The minimum size of an n‐vertex H‐saturated graph is denoted by sat(n,H). We prove sat(n,Ck)=n+n/k+O((n/k2)+k2)holds for all n≥k≥3, where Ck is a cycle with length k. A graph G is H‐semisaturated if G+e contains more copies of H than G does for ∀e∈E(G¯). Let ssat (n,H) be the minimum size of an n‐vertex H‐semisaturated graph. We have ssat(n,Ck)=n+n/(2k)+O((n/k2)+k).We conjecture that our constructions are optimal for n>n0(k). © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013

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An $r$-matching in a graph $G$ is a collection of edges in $G$ such that the distance between any two edges is at least $r$. This generalizes both matchings and induced matchings as matchings are $1$-matchings and induced matchings are $2$-matchings. In this paper, we study the minimum and maximum number of $r$-matchings in a tree with fixed order.