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Largest number of maximal cographical node-induced subgraphs of an n-node graph.
1

%I #8 Mar 15 2022 05:23:29

%S 1,1,1,4,10,12,23,38,64

%N Largest number of maximal cographical node-induced subgraphs of an n-node graph.

%C This sequence is log-superadditive, i.e., a(m+n) >= a(m)*a(n). By Fekete's subadditive lemma, it follows that the limit of a(n)^(1/n) exists and equals the supremum of a(n)^(1/n).

%F a(m+n) >= a(m)*a(n).

%F Limit_{n->oo} a(n)^(1/n) >= 64^(1/9) = 1.58740... .

%e All graphs with at most three nodes are cographs, so a(n) = 1 for n <= 3 and any graph is optimal.

%e All optimal graphs (i.e., graphs that have n nodes and a(n) maximal cographical subgraphs) for 4 <= n <= 9 are listed below. Since a graph is a cograph if and only if its complement is a cograph, the optimal graphs come in complementary pairs.

%e n = 4: the path of length 3 (self-complementary);

%e n = 5: the 5-cycle (self-complementary);

%e n = 6: the Hajós graph (also known as a Sierpiński sieve graph) and its complement;

%e n = 7: the elongated triangular pyramid and its complement;

%e n = 8: the Möbius ladder and its complement (the antiprism graph);

%e n = 9: the pentagonal bipyramid with an additional path of length 3 between the two apex nodes (self-complementary).

%Y Cf. A000084, A000669.

%Y For a list of related sequences, see cross-references in A342211.

%K nonn,more

%O 1,4

%A _Pontus von Brömssen_, Mar 08 2022