On strong (weak) independent sets and vertex coverings of a graph Original Research Article

A vertex image in a graph image is strong (weak) if image image for every u adjacent to image in G. A set image is said to be strong (weak) if every vertex in S is a strong (weak) vertex in G. A strong (weak) set which is independent is called a strong independent set [SIS] (weak independent set [WIS]). The strong (weak) independence number image is the maximum cardinality of an SIS (WIS). For an edge image, image strongly covers the edge x if image in G. Then u weakly covers x. A set image is a strong vertex cover [SVC] (weak vertex cover [WVC]) if every edge in G is strongly (weakly) covered by some vertex in image. The strong (weak) vertex covering number image image is the minimum cardinality of an SVC (WVC). In this paper, we investigate some relationships among these four new parameters. For any graph G without isolated vertices, we show that the following inequality chains hold: image and image. Analogous to Gallaiʹs theorem, we prove image and image. Further, we show that image and image and find a necessary and sufficient condition to attain the upper bound, characterizing the graphs which attain these bounds. Several Nordhaus–Gaddum-type results and a Vizing-type result are also established.