introdution to Graph theory

Question

I recently had my midterm and this was one of the questions I could not tackle upon I'm just curious to know how would the problem be solved

Answer 1

An subgraph of a graph $G$ is any graph whose vertices and edges are a subset of those in $G$. In your example, $H=V_h,E_h$ where $V_h={x,y}$ and $E_H={{xy}}$ is one example.

An induced subgraph of $G$ is the same as a subgraph but must contain all of the edges in $G$ between vertices in the subset. The $H$ above is also an example of an induced subgraph.

A labelled subgraph means that even if two subgraphs of $G$ look the same (for example, if they both contain only two adjacent vertices), if they contain any different vertices or different edges from each other, then they are considered different. In other words, if two subgraphs are non-isomorphic, then we count them as two labelled subgraphs (the labels of the vertices do not matter). Thus the subgraph of $G$ containing $x$ and $y$ and the edge $xy$ and the subgraph containing $y$ and $z$ and the edge $yz$ are counted as two different labelled subgraphs.