[Math] Definition of induced cycle

Question

According to Diestel (page 4):
"If $G' \subseteq G$ and $G'$ contains all the edges $xy \in E$ with $x, y \in V'$, then $G'$ is an induced subgraph of $G$"

According to Wikipedia
"induced cycle is a cycle that is an induced subgraph of $G$; induced cycles are also called cordless cycles "

Does the definition by Diestel imply induced cycles are chordless?

In this graph, does induced subgraph $G[\{a,b,c,d\}]$ include edge
$ac$?

Both $a$ and $c$ are in $V'$.

Answer 1

Shortly, it does.

It is easier to think of induced subgraphs in terms of vertices: you take any $V' \subseteq V$ and take all the edges from $E$ that make sense, that is, both their ends are in $V'$. In fact, $(a,c) \in E$ and $a,c \in V'$ implies $(a,c) \in E'$.

I guess your confusion might be caused by the term "cycle" which has here two similar meanings:

1. A closed path in a graph, i.e. a sequence $C = (c_0,\ldots,c_{n-1}) \subseteq V$ such that $(c_{i},c_{i+1}) \in E$ and $(c_{n-1},c_0) \in E$. Note that here, there may be many chords inside, and $C$ would still be a cycle.
2. A graph that is a cycle. There are no other edges, in fact it is a connected 2-regular graph (i.e. connected and $\forall v.\ \mathrm{deg}(v) = 2$).

So a cycle-1 is chordless if and only if it is an induced cycle-2.

I hope it explained something ;-)