On the Planarity of Generalized Line Graphs

One of the most familiar derived graphs is the line graph. The line graph L(G) of a graph G is that graph whose vertices are the edges of G where two vertices of L(G) are adjacent if the corresponding edges are adjacent in G. Two nontrivial paths P and Q in a graph G are said to be adjacent paths in G if P and Q have exactly one vertex in common and this vertex is an end-vertex of both P and Q. For an integer l ≥ 2, the l-line graph Ll(G) of a graph G is the graph whose vertex set is the set of all l-paths (paths of order l) of G where two vertices of Ll(G) are adjacent if they are adjacent l-paths in G. Since the 2-line graph is the line graph L(G) for every graph G, this is a generalization of line graphs. In this work, we study planar and outerplanar properties of the 3-line graph of connected graphs and present characterizations of those trees having a planar or outerplanar 3-line graph by means of forbidden subtrees.