Packing 3-vertex paths in claw-free graphs and related topics Original Research Article

A image-factor of a graph image is a spanning subgraph of image whose every component is a 3-vertex path. Let image be the number of vertices of image and image the domination number of image. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it does not have an induced subgraph isomorphic to a claw. Our results include the following. Let image be a 3-connected claw-free graph, image, image, and image a 3-vertex path in image. Then image if image, then image has a image-factor containing (avoiding) image, image if image, then image has a image-factor, image if image, then image has a image-factor, image if image and image is either cubic or 4-connected, then image has a image-factor, image if image is cubic with image and image is a set of three edges in image, then image has a image-factor if and only if the subgraph induced by image in image is not a claw and not a triangle, image if image, then image has a image-factor for every vertex image and every edge image in image, image if image, then there exist a 4-vertex path image and a claw image in image such that image and image have image-factors, and imageimage and if in addition image is not a cycle and image, then image. We also explore the relations between packing problems of a graph and its line graph to obtain some results on different types of packings and discuss relations between image-packing and domination problems.