On Strong (weak) Independent Sets and Vertex Coverings of a Graph

Kamath , S S and Bhat, R S (2007) On Strong (weak) Independent Sets and Vertex Coverings of a Graph. Discrete Mathematics, 307 (9-10). pp. 1136-1145. ISSN 0012-365X

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## Abstract

A vertex v in a graphG=(V ,E) is strong (weak) if deg(v) deg(u) (deg(v) deg(u)) for every u adjacent to v in G.A set S ⊆ V 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 s = s(G)(w = w(G)) is the maximum cardinality of an SIS (WIS). For an edge x =uv, v strongly covers the edge x if deg(v) deg(u) in G. Then u weakly covers x. A set S ⊆ V is a strong vertex cover [SVC] (weak vertex cover [WVC]) if every edge in G is strongly (weakly) covered by some vertex in S. The strong (weak) vertex covering number s = s(G) (w = w(G)) 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: ssw and sww. Analogous to Gallai’s theorem, we prove s+w=p and w+s=p. Further, we show that sp− and wp− 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.

Item Type: Article © 2006 Published by Elsevier B.V. Strong (weak) vertices; Strong (weak) vertex cover; Strong (weak) independent sets Engineering > MIT Manipal > Mathematics MIT Library 19 Oct 2011 09:33 19 Oct 2011 09:33 http://eprints.manipal.edu/id/eprint/1344