  Let $G$ be a graph on $n$ vertices. A vertex of $G$ with degree at least
$n/2$ is called a heavy vertex, and a cycle of $G$ which contains all the heavy
vertices of $G$ is called a heavy cycle. In this paper, we characterize the
graphs which contain no heavy cycles. For a given graph $H$, we say that $G$ is
$H$-\emph{heavy} if every induced subgraph of $G$ isomorphic to $H$ contains
two nonadjacent vertices with degree sum at least $n$. We find all the
connected graphs $S$ such that a 2-connected graph $G$ being $S$-heavy implies
any longest cycle of $G$ is a heavy cycle.
