Independent sets in {claw,K4}-free 4-regular graphs

We prove that each {claw, K4}-free 4-regular graph, with just one class of exceptions, is a line graph. Applying this result, we present lower bounds on the independence numbers for {claw, K4}-free 4-regular graphs and for {claw, diamond}-free 4-regular graphs. Furthermore, we characterize the extremal graphs attaining the bounds.