Why is 2-connectedness characterized by the existence of an ear decomposition ? Reading the book "Introduction to Graph Theory" (West, 2nd ed.) I have come across the following definition and statement:



Why is this statement true ?

Suppose I have a graph $G$ that consist of a cycle $C$ and a path $P$ that share no vertices. Then $G$ has an ear decomposition ? Well, I can decompose $G$ into $P_o = C$ and $P_1 = P$ ? $P_o$ is a cycle and $P$ is an ear of $P_0 \cup P_1$ - $P$ is a maximal path whose interval vertices have degree $2$ in $G$ ? But $G$ is not connected and therefore is $0$-connected and not $2$-connected ?
"Ear decomposition" is not a term I've heard before but it's incorrectly defined, as you've observed, since it doesn't require that the graph be connected, let-alone 2-connected.  An "ear" needs to be defined as having each endpoint adjacent to at least one further vertex of&nbsp;$G$.  (Actually, each endpoint would have to be adjacent to at least two further vertices, or the path wouldn't be maximal.)

With the correct definition, the theorem is true.