Self-complementary graph

Self-complementary graph

A self-complementary graph is a graph which is isomorphic to its complement. The simplest self-complementary graphs are the 4-vertex path graph and the 5-vertex cycle graph.

Self-complementary graphs are interesting in their relation to the graph isomorphism problem: the problems of checking whether two self-complementary graphs are isomorphic and of checking whether a given graph is self-complementary are polynomial-time equivalent to the general graph isomorphism problem. [Colbourn M.J., Colbourn Ch.J. "Graph isomorphism and self-complementary graphs", "SIGACT News", 1978, vol. 10, no. 1, 25-29]

An "n"-vertex self-complementary graph has exactly half number of edges of the complete graph, i.e., "n"("n" − 1)/4 edges, and (if there is more than one vertex) it must have diameter either 2 or 3. [Sachs, H. (1962) "Über selbstkomplementäre Graphen." "Publ. Math. Debrecen" vol. 9, 270-288] Since "n"("n" −1) must be divisible by 4, "n" must be congruent to 0 or 1 mod 4; for instance, a 6-vertex graph cannot be self-complementary.

Every Paley graph is self-complementary.

References


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