On the Planarity of G
Liu Xiao-ping · Shinjang dashösi ilmiy jurnili · 2006
Let $G$ be a simple graph. The total graph $G+{+++}$ is the graph with vertex set $V(G)∪E(G)$, in which the vertex $x$ and $y$ are joined by an edge if one of the following conditions holds: (i) $x,y∈V(G)$,and $x$ and $y$ are adjacent in $G$, (ii) $x, y∈E(G)$, and $x$ and $y$ are adjacent in $G$, (iii) one of $x$ and $y$ is in $V(G)$and the other is in $E(G)$, and they are incident in $G$.$G+{---}$ is the complement of $G+{+++}$. In this paper, it is shown $G+{---}$ is planar if and only if either $｜V(G)｜≤3$ or $G$ is isomorphic to one of the following graphs: $2K-2, C-4,K-4-e, K-4, 2K-1+ K-3, K-{1,4}, K-1+K-{1,3}, 2K-1+ P-3$.