On matching and total domination in graphs

A set image of edges of a graph image is a matching if no two edges in image are incident to the same vertex. A set image of vertices in image is a total dominating set of image if every vertex of image is adjacent to some vertex in image. The matching number is the maximum cardinality of a matching of image, while the total domination number of image is the minimum cardinality of a total dominating set of image. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every image-regular graph with image has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.