The total irredundance numbers on graphs

Abstract

This paper discusses the total irredundance relations between the graph G and its clone-contraction graph H, that is, let H be the clone-contraction graph of G and v1, v2, ⋯ vk be all contraction vertices of H. If S is a maximal total irredundant set of H such that A = S∩{v1, v2, ⋯, vk} contains as few vertices as possible, then S′ = S − A is the maximal total irredundant set of G. Furthermore, we obtain the bound of the total irredundance number: irt ⩽ ΔG / 2Δ(G)+1n, which n is the order of graph G, and Δ(G) is maximum degree in G.