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1976•Journal of the Australian Mathematical SocietyOpen access

Stability of line graphs

Douglas D. Grant

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Abstract

Abstract We present in this paper a discussion on some stability properties of line graphs. After relating the semi-stability properties of the line graph of a graph to a concept of Sheehan, we proceed to deduce that, with fully characterised lists of exceptions, the line graphs of trees and unicyclic graphs are semi-stable. We then discuss the problem of deciding which line graphs are stable. Via a discovery of the finite number of graphs G such that both G and its complement have stable line graphs, we show that P4 is the only self-complementary graph whose line graph is stable.

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Abstract We present in this paper a discussion on some stability properties of line graphs. After relating the semi-stability properties of the line graph of a graph to a concept of Sheehan, we proceed to deduce that, with fully characterised lists of exceptions, the line graphs of trees and unicyclic graphs are semi-stable. We then discuss the problem of deciding which line graphs are stable. Via a discovery of the finite number of graphs G such that both G and its complement have stable line graphs, we show that P4 is the only self-complementary graph whose line graph is stable.

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Available abstract

Abstract We present in this paper a discussion on some stability properties of line graphs. After relating the semi-stability properties of the line graph of a graph to a concept of Sheehan, we proceed to deduce that, with fully characterised lists of exceptions, the line graphs of trees and unicyclic graphs are semi-stable. We then discuss the problem of deciding which line graphs are stable. Via a discovery of the finite number of graphs G such that both G and its complement have stable line graphs, we show that P4 is the only self-complementary graph whose line graph is stable.

Key concepts: Line graph, Mathematics, Combinatorics, Block graph, Pathwidth, Cograph, 1-planar graph, Complement (music)

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