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$F$-Continuous Graphs

For a nontrivial connected graph $F$, the $F$-degree of a vertex $v$ in
a graph $G$ is the number of copies of $F$ in $G$ containing $v$. A graph
$G$
is $F$-continuous (or $F$-degree continuous) if the $F$-degrees of every
two adjacent vertices of $G$ differ by at most 1. All $P_3$-continuous
graphs
are determined. It is observed that if $G$ is a nontrivial
connected graph that is $F$-continuous for all nontrivial connected
graphs $F$, then either $G$ is regular or $G$ is a path. In the
case of a 2-connected graph $F$, however, there always
exists a regular graph that is not $F$-continuous. It is also shown that for
every
graph $H$ and every 2-connected graph $F$, there exists an
$F$-continuous graph $G$ containing $H$ as an induced subgraph.