$OC$-Irredundance, $CO$-Irredundance and Maximum Degree in Trees

Abstract

A vertex subset $X$ of a simple graph is called OC-irredundant (respectively CO-irredundant) if for each $v\in X$, $N(v)–N[X–\{v\}]\ne \mathrm{\varnothing }$ (respectively $N[v]–N(X–\{v\})\ne \mathrm{\varnothing }$). Sharp bounds involving order and maximum degree for the minimum cardinality of a maximal OC-irredundant set and a maximal CO-irredundant set of a tree are obtained, and extremal trees are exhibited.