New results and open problems in line graphs

Abstract

AbstractGiven a graph G with at least one edge, the line graph L(G) is that graph whose vertices are the edges of G, with two of these vertices being adjacent if the corresponding edges are adjacent in G. The line graph transformation is one of the most extensively studied, and the concept extends naturally to digraphs. The authors have recently published a book Line Graphs and Line Digraphs that covers many properties and generalizations of both types of structure. In this paper we discuss some recent progress in this area. We include a discussion of several recognition algorithms related to line graphs, and some new results on the line completion numbers. We also present some open problems and directions for further research.

Keywords

• Line graph
• super line graph
• line completion number
• line graph recognition
• 05C76
• 05C75