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A graph and its complement with specified properties V: The self-complement index

Part of: Graph theory

Published online by Cambridge University Press:  26 February 2010

Jin Akiyama ,
Geoffrey Exoo  and
Frank Harary
Jin Akiyama
Affiliation:
University of Michigan, Department of Mathematics, Ann Arbor, M1 48109, U.S.A.
Geoffrey Exoo
Affiliation:
University of Michigan, Department of Mathematics, Ann Arbor, M1 48109, U.S.A.
Frank Harary
Affiliation:
University of Michigan, Department of Mathematics, Ann Arbor, M1 48109, U.S.A.
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Abstract

The self-complement index s(G) of a graph G is the maximum order of an induced subgraph of G whose complement is also induced in G. This new graphical invariant provides a measure of how close a given graph is to being selfcomplementary. We establish the existence of graphs G of order p having s(G) = n for all positive integers n < p. We determine s(G) for several families of graphs and find in particular that when G is a tree, s(G) = 4 unless G is a star for which s(G) = 2.

MSC classification

Secondary: 05C99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 27 , Issue 1 , June 1980 , pp. 64 - 68
Copyright
Copyright © University College London 1980

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References

1.Akiyama, J. and Harary, F.. “A graph and its complement with specified properties III: Girth and circumference”, Int'l J. Math, and Math. Sci., 2 (1979), 685–692.CrossRefGoogle Scholar
2.Akiyama, J. and Harary, F.. “A graph and its complement with specified properties IV: Counting selfcomplementary blocks”, J. Graph Theory, to appear.Google Scholar
3.Harary, F.. Graph theory (Addison-Wesley, Reading, Massachusetts, 1969).CrossRefGoogle Scholar
4.Harary, F. and Read, R. C.. “Is the null graph a pointless concept?” Graphs and combinatorics (Bari, R. and Harary, F., eds.), Springer Lecture Notes, 406 (1974), 37–44.CrossRefGoogle Scholar