4-regular claw-free IM-extendable graphs

A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic to image. The kth power of a graph G, denoted by image, is the graph with vertex set image in which two vertices are adjacent if and only if the distance between them in G is at most k. In this paper, the 4-regular claw-free IM-extendable graphs are characterized. It is shown that the only 4-regular claw-free connected IM-extendable graphs are image, image and image, image, where image is the graph with image vertices image, image, such that for each i with image, image is a clique of image and image. We also show that a 4-regular strongly IM-extendable graph must be claw-free. As a consequence, the only 4-regular strongly IM-extendable graphs are image, image and image.