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Super Line Graphs and Super Line Digraphs

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Line Graphs and Line Digraphs

Part of the book series: Developments in Mathematics ((DEVM,volume 68))

Abstract

Instead of simply taking, from a given graph G, one edge at a time to form the line graph, one could take pairs of edges as the vertices and let two of these vertices be adjacent if at least one of the edges in one set is adjacent in G to at least one of those in the other set. This new graph is called a line graph of index 2. Naturally, line graphs of index k are defined analogously, and they constitute the subject of this chapter and have the name super line graphs. Subgraphs, independent sets, and paths and cycles are among the aspects of these objects that are discussed in subsequent sections. The case k = 2 is the examined in greater detail. We then turn to what we call the ‘line-completion number’ of a graph G, which is the least index r for which the super line graph is complete. This turns out to be particularly interesting as a purely number-theoretical problem that is difficult to compute or even approximate in many cases. We then conclude the chapter with sections on super line digraphs and super line multigraphs.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne, IN, USA

    Lowell W. Beineke

  2. Department of Computer Science, Ball State University, Muncie, IN, USA

    Jay S. Bagga

Authors
  1. Lowell W. Beineke
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  2. Jay S. Bagga
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Beineke, L.W., Bagga, J.S. (2021). Super Line Graphs and Super Line Digraphs. In: Line Graphs and Line Digraphs. Developments in Mathematics, vol 68. Springer, Cham. https://doi.org/10.1007/978-3-030-81386-4_15

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