On Stability Number of Upper Irredundance Number in Graphs

A vertex v in a vertex-subset I of an undirected graph G is said to be redundant if its closed neighborhood is contained in the union of closed neighborhoods of vertices of I − {v}. In the context of a communication network , this means that any vertex that may receive communications from I may also be informed from I − {v} . The irredundance number ir(G) is the minimum cardinality taken over all maximal sets of vertices having no redundancies, while the upper irredundance number is the maximum cardinality of any such set . In this paper, we investigate SN(G), the stability number of upper irredundance number, i.e., the maximum cardinality among all sets of edges E′ for which IR(G− E′) = IR(G). For a nonempty connected graph G with order n(n ≥ 2), we show that SN(G) ≤ n − 2 . Moreover , we obtain SN(G) ≤ (IR(G)− 1)∆(G)− 1 when IR(G) ≥ 2.