TITLE: Perfect matchings, blossoms, and biconnected edges: need help understanding a claim
QUESTION [5 upvotes]: I am reading an early paper by Deming on Koenig graphs 1, and I am stuck trying to understand a claim in the paper. 
Preliminaries
Let $G$ be a (finite, undirected, simple) graph. Fix a maximum matching $M$ of $G$. The edges in $M$ are said to be heavy; those not in $M$ are said to be light. A vertex $v$ of $G$ is said to be exposed relative to $M$ if $v$ is not the endpoint of any heavy edge of $G$. An alternating path from vertex $u$ to vertex $v$ is a path $p(u,v)$ whose edges are alternately light and heavy; such a path may begin or end with either a light edge or a heavy edge. 
An odd cycle $v_{0},v_{1},\dotsc,v_{2k},v_{0}\;;\;k\geq1$ is called a blossom if edges $v_{1}v_{2},v_{3}v_{4},\dotsc,v_{2k-1}v_{2k}$ are heavy. The vertex $v_{0}$ in such a blossom is called the blossom tip. 
Suppose there is an alternating path $v_{1},v_{2},\dotsc,v_{2k}\;;\;k\geq1$ in $G$ where the edges $v_{1}v_{2}$ and $v_{2k-1}v_{2k}$ are heavy and the vertices $v_{1}$ and $v_{2k}$ are blossom tips. "A configuration of this type" is called a blossom pair of $G$. 
(I add the quotes because it is not clear to me what the author means by a configuration of this type: Does he mean just the path $v_{1},v_{2},\dotsc,v_{2k}$, or does he also mean to include the blossoms whose tips are $v_{1}$ and $v_{2k}$?)
The claim
Let $G$ be a graph with a perfect matching $M$, and $x_{1}x_{2}$ and $y_{1}y_{2}$ be heavy edges which belong to a blossom pair $B$ with respect to the matching $M$. Then for each $i,j=1,2$, there is an alternating path $p(x_{i},y_{j})$ beginning and ending with light edges and contained in $B$. 
My question
Consider the graph below. It satisfies the conditions of the claim: It has a perfect matching (indicated by the "heavy" edges) and contains/is a blossom pair (depending on how we interpret the definition of a blossom pair above) with respect to this matching. Now $x_{1}x_{2}$ and $y_{1}y_{2}$ are heavy edges which belong to the blossom pair (for either interpretation), but there is no alternating path $p(x_{1},y_{2})$ which begins and ends with light edges. What gives?

Deming quotes Edmonds' Paths, trees, and flowers paper for terminology up to and including blossom tip. Just to be sure, I checked this latter paper: Edmonds uses "path" to mean "simple path", so it cannot be that Deming means "alternating walk" here when he says "alternating path". 
What am I missing here?
Additional information
(May be of help in resolving my question.)
Deming goes on to define a pair of heavy edges $x_{1}x_{2}$ and $y_{1}y_{2}$ to be biconnected if there are alternating paths $p(x_{i},y_{j})$ for $i=1,2\;;\;j=1,2$ which begin and end with light edges. Then he states that any pair of biconnected heavy edges is contained in a blossom pair of the graph. 
Picture mentioned in the comment to SE318's answer

1 Independence Numbers of Graphs -- An Extension of the Koenig-Egervary Theorem. Robert W. Deming, Discrete Mathematics 27 (1979) 23-33. PDF

REPLY [2 votes]: I tried to use the claim about biconnected pairs of heavy edges to clarify
what must be included in the "blossom pair."
But it seems problematic!

In this picture, $x_1 x_2$ and $y_1 y_2$ are a biconnected pair of heavy edges (right?).
Yet, they are contained in no blossom pair.
The only blossoms are:

$a,x_1,x_2,a$
$b_,x_1,x_2,b$
$x_1, a, b, x_1$, and 
$x_2, a, b, x_2$.

Any alternating path, starting and ending with heavy edges, between any of these blossom tips, does not involve $y_1 y_2$. Hence that edge is not involved in any blossom pair in this graph, for any of the potential definitions.
So, I believe the claims as stated are wrong. However, what the author wants to use is this:

A graph contains a biconnected pair of heavy edges if and only if it
  also contains a blossom pair.

I believe that this is true, despite the erroneous statements attempting to prove it.
In particular, in the claim in the question, what the author actually requires is:

Let $G$ be a graph with a perfect matching $M$, and $B$ be a blossom pair with respect to the matching $M$.
  Then there exist a pair of heavy edges $x_1x_2$ and $y_1y_2$ belonging to $B$
  such that for each $i,j=1,2$, there is an alternating path $p(x_i,y_j)$ beginning
  and ending with light edges and contained in $B$.