Question about notation

Question

I would like to know what $G\cup \left\{v \right\}$ means. Does it mean to tack on another vertex to the graph $G$ by placing all possible edges between $v$ and the vertices of $G$?

Answer 1

So I assume that $G$ is a graph? Which in your text may defined tobe a tuple $(V,E,I)$, where $V$ is the set of vertices, $E$ the set of edges and $I$ te incidence relation ...? In that case $G$ is not a set (at first sight) and special notation such as $G\cup \{v\}$ neds to be defined in your text.

It is in principle possible to define $G$ as the set of its vertices (urelements) and edges (sets containing two urelements) so thet $\in$ plays the role of incidence relation; but I have never seen an introductory text going this way. In that case, $G\cup \{v\}$ is of course the graph obtained from $G$ by adding a new verte $v$ withoiut any edges (or it is $G$ itself if already $v\in G$).