A Note on the Neighbor Condition for Up-Embedidability of Graphs

Abstract

Different neighbor conditions are considered in \([3,4,9]\) for a graph up-embeddable. In this paper, we consider the neighbor conditions of all the pairs of vertices with diameter \(2\) and obtain the following new result: if \(|N_G(u) \cap N_G(v)| \geq 2\) for any two vertices \(u,v \in D\) where \(D = \{(u, v) | d_G(u, v) = 2, u,v \in V(G)\}\), then \(G\) is up-embeddable.