Perfect matchings, blossoms, and biconnected edges: need help understanding a claim 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
 A: Instead of "blossom pair" Sterboul defined an "$M$-posy". Here's the passage from Schrijver:


To characterize graphs $G$ with $ν(G) = τ (G)$, Sterboul defined, for any
  graph $G = (V,E)$ and any matching $M$ in $G$, an $M$-posy to be an even-length
  $M$-alternating closed walk $(v_0, v_1, . . . , v_t)$, with $v_{i−1}v_i ∈ M$ if $i$ is
  even, such that there exist $i < j$ with $i$ odd and $j$ even, $v_1, . . . , v_j$ all distinct, $v_{j+1}, . . . , v_t$ all distinct, and
(30.54) $v_i = v_t,$ $v_{i+1} = v_{t−1}, . . . , v_j = v_{t+i−j}$ .
Lemma 30.13α. If there exists an even-length $M$-alternating closed walk
  $C = (v_0, v_1, . . . , v_t)$ with $v_i = v_j$ for $i, j$ of different parity, then there exists an $M$-posy.

Both things you have drawn are $M$-posys. As you can see it's rather hairy to write down a formal definition fully capturing the picture.

EDIT: While I still haven't figured out what exactly Deming means by "biconnected" edges, his ultimate result in that section, his theorem 2, sheds some light on what he's trying to say:

Theorem 2. An arbitrary graph G is K-E iff for any maximum matching M, the extension G’ contains no pair of biconnected heavy edges.

His definition of extension graph is straight-forward (and illustrated with examples so there's little doubt what it means):

To construct $G’$, let $X$ be the set of exposed vertices of $G$ relative to a maximum
  matching $M$. For each $x \in X$, add a new vertex $x’$ and new edges $xx’$ and $\{x’y : y \text{ is adjacent to } x\}$. $M’ = M \cup \{xx’ : x \in X\}$ is a perfect matching for $G’$.

(Exposed vertex means not included in the match.)
It helps to compare Deming's theorem 2 with its equivalent theorem 30.13 in Schrijver (which I'm abbreviating slightly):

G has the Koenig property iff for each maximum-size matching M there is no M-flower and no M-posy

where M-flower is defined as the one-sided version of the M-posy:

If you take an M-flower and apply Deming's extension construction to it, you get an M-posy; only $v_0$ gets duplicated to a $v_0'$. So you can also phrase the theorem as: G has the Koenig property iff for each maximum-size matching of G' (denoted by M') there is no M'-posy. In other words, the forboding/constraint about M-flowers and M-posys in the original graph can be expressed just a constraint about  M-posys in Deming's extenstion graph. So by his "no pair of biconnected heavy edges" in G' he most likely means there's no M'-posy in G'. It's still not clear to me what exactly Deming means by "biconnected heavy edges" etc.

EDIT2: Actually, I understand now what Deming means by "biconnected heavy edges": there must be four paths beginning and ending with light edges: $x_1y_1$, $x_1y_2$, $x_2y_1$ and $x_2y_2$ must all be paths with this property (but the four paths need not be entirely distinct from each other). This def/interpretation is actually equivalent to the two heavy edges belonging to the opposite blossom ends of an M-possy. With this interpretation the "biconnected heavy edges" can't be in the same blossom (you'd have two light edges in a row by following the blossom cycle; and if you go around the other blossom it's not a simple path anymore), nor can they be in the "bridge" between the blossoms (you won't have four simple paths either then). So in the two graphs/matchings you've drawn, the highlighted $x_1x_2$ and $y_1y_2$ do not actually meet Deming's def of "biconnected heavy edges".
And getting back to "the claim", since it was intended to be half of this equivalence between "biconnected heavy edges" and M-posy, it should have been stated [by Deming] as: Let $G$ be a graph with a perfect matching $M$, and $x_{1}x_{2}$ and $y_{1}y_{2}$ be heavy edges each belonging to an opposite blossom of 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 $B$. 

Note however that the equivalence between M-posy and biconnected heavy edges is wrong, because it fails in the other direction; after "the claim" [which at least could be patched] the paper says "A straightforward argument shows that a pair of biconnected heavy edges $x_1x_2$ and $y_1y_2$, are contained in a blossom pair B." Kundor shows that you can have biconnected heavy edges but one (or both) of the pair need not belong to an M-posy; this is because one (or both) of the biconnected heavy edges can belong to an even-length cycle in the matching and this cycle can share a heavy edge with an odd-length cycle (i.e. actual blossom belonging to an M-posy) which in turn can share another heavy edge with another even-length cycle. So you get some "extra" pairs of biconnected heavy edges not beloning to the actual M-posy but whose alternating, four go-between paths "pass through" the M-posy. (Without the M-posy facilitating this "crossover" you wouldn't get the 4 paths though.) Here's somewhat crude picture of Kundor's idea in a more general setting:

Deming's Theorem 2 still is correct though, but because the existence of a pair of biconnected heavy edges implies the existence of at least one M-posy (in the matching), even though not all biconnected heavy edges need belong to an M-posy. Proof that the existence of a pair of biconnected heavy edges implies the existence of an M-posy is not quite that trivial/obvious though.
A: I could be wrong here, but one way to make sense of this... I think the blossom pair is literally just referring to the two blossoms in question(the ones in which $v_1$ and $v_{2k}$ are blossom tips), and NOT the path in between, so your edges $x_1x_2$ and $y_1y_2$ are not in the blossom pair because they are in the path, not the blossoms.
A: 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 $B$.

