A graph property is called hereditary if it closed with respect to deleting vertices (i.e., all induced subgraphs inherit the property). A graph property is called additive if it is closed with respect to taking disjoint unions.
It is not hard to find properties that are hereditary, but not additive. Two simple examples:
$\;\;\;$ (1) The graph is complete.
$\;\;\;$ (2) The graph does not contain two vertex-disjoint cycles.
In these cases it is obvious that the property is inherited by induced subgraphs, but taking two disjoint graphs that have the property, their union may not preserve it.
Both of the above examples are polytime decidable properties (although for (2) it is somewhat less trivial). If we want harder properties, they could still be created by following the pattern of (2), but replacing the cycles with more complicated graph types. Then, however, we can easily run into the situation where the problem does not even remain in $NP$, under standard complexity assumptions, such as $NP\neq coNP$. It appears less trivial to find an example which stays within $NP$, but it is still hard.
Question: Do you know a (preferably natural) $NP$-complete graph property that is hereditary, but not additive?