The irredundance number and maximum degree of a graph


A vertex η in a subset X of vertices of an undirected graph is redundant if its closed neighborhood is contained in the union of closed neighborhoods of vertices of X-{η}. In the context of a communications network, this means that any vertex that may receive communications from X may also be informed from X-{η}. The irredundance number ir(G) is the minimum cardinality taken over all maximal sets of vertices having no redundancies. In this note we show that ir(G) ≥ n/(2Δ-1) for a graph G having n vertices and maximum degree Δ. © 1984.

Publication Title

Discrete Mathematics