On Generalized Line Graphs with Crossing Number One

Hoffman (1968) introduced the notion of generalized line graphs. In 1985, Syslo and Topp studied which of these graphs have crossing number zero. For a nontrivial connected graph G, let L(G, v) denote the complete graph on the set of all edges incident to vertex v of G. For a function f : V(G) → N∗, the set of non-negative integers, let {CP(f(v)) : v ∈ V(G)} be the family of cocktail party graphs disjoint from each other and from G as well as L(G). The generalized line graph L(G,f) of G is the graph: uv∈V(G) {L(G,v) + CP(f(v))}. It is easy to see that L(G,f) = L(G) if and only if f(v) = 0 for every v ∈ V(G). In this paper we obtain a necessary and sufficient condition for a generalized line graph to have crossing number 1. Our result gives a natural generalization of that for line graphs given by Kulli et al. (1979).