On the 3-γt-Critical Graphs of Order Δ(G) + 3

Abstract

Let γt(G) be the total domination number of graph G, a graph G is k-total domination vertex critical (or just k-γt-critical) if γt (G) = k, and for any vertex v of G that is not adjacent to a vertex of degree one, γt(G - v) = k - 1. Mojdeh and Rad [6] proposed an open problem: Does there exist a 3-γt-critical graph G of order Δ(G) + 3 with Δ(G) odd? In this paper, we prove that there exists a 3-γtcritical graph G of order Δ(G) + 3 with odd Δ(G) ≥ 9.