Characterizing 2image-critical graphs and image-extendable graphs

Let image be a graph with even order. Let image be a matching in image and image, be the image-unsaturated vertices in image. Then image has a perfect matching if and only if there are image independent image-augmenting paths joining the image vertices in pairs. Let image be a graph with a perfect matching image. It is proved that image is image-critical if and only if for any image vertices image in image, there are image independent image-alternating paths image joining the image vertices in pairs such that image start and end with edges in image. It is also proved that image is image-extendable if and only if, for each image with image and each image with image, and for any image pairs of image-alternating paths image with image, image in image, there exist independent image-alternating paths image in image joining the vertices in image, where image, which start and end with edges in image.