On the Planarity of Generalized Line Graphs

Authors

Khawlah H. Alhulwah, Western Michigan UniversityFollow
Mohra Zayed, Western Michigan UniversityFollow
Ping Zhang, Western Michigan UniversityFollow

Publication Date

2018

Abstract

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.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Alhulwah, Khawlah H.; Zayed, Mohra; and Zhang, Ping (2018) "On the Planarity of Generalized Line Graphs," Theory and Applications of Graphs: Vol. 6: Iss. 1, Article 2.
DOI: 10.20429/tag.2019.060102
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol6/iss1/2