I met the following two combinatorial concepts during a study outside of combinatorics. I am wondering if there are common terminologies in combinatorics. 


1. A finite graph $G$ has the following property:



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> For any vertex $a$ and edge $\{b,c\}$ of $G$, there is an edge connecting them: there is at least one of $\{a,b\}$ or $\{a,c\}$ in $G$.
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2. Let $P = \lambda\_1 \ge \lambda\_2 \ge \cdots \ge \lambda\_k$ be an integer partition of $n$. Suppose that $P$ has the following property:



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> If there are two index sets $I$ and $J$ such that $\sum\_{i \in I}\lambda\_i = \sum\_{j \in J}\lambda\_j$, then the restricted partitions $\{\lambda\_i\}\_{i \in I}$ and $\{\lambda\_j\}\_{j \in J}$ are same. 
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For instance, $5 \ge 3 \ge 2 \ge 2$ does not have the property because $2+3=5$, but $5 \ge 5 \ge 3 \ge 3$ has the property.


Are there common names for these two properties?



Terminology in combinatorics
This is only an answer to your first question. 


The graphs with your property can be characterised by forbidding as induced subgraph $\overline{P}\_3$. By Andreas Blass's comment, your property implies that the complement of the graph is a disjoint union of cliques. These graphs are called complete multipartite graphs. 


Sometimes for this kind of question the site <http://graphclasses.org/> can be useful.


