A342211 Largest number of maximal acyclic node-induced subgraphs of an n-node graph.

1, 1, 3, 6, 10, 15, 22, 38, 64

Offset: 1

Author

Pontus von Brömssen, Mar 05 2021

Keywords

• nonn
• more

Comments

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). - Pontus von Brömssen, Mar 03 2022

a(10) >= 105.

Examples

			All optimal graphs (i.e., graphs having n nodes and a(n) maximal acyclic subgraphs) for 1 <= n <= 9 are listed below. Here, FCB(n_1, ..., n_k) denotes the full cyclic braid graph with cluster sizes n_1, ..., n_k, as defined by Morrison and Scott (2017), i.e., the graph obtained by arranging complete graphs of orders n_1, ..., n_k (in that order) in a cycle, and joining all pairs of nodes in neighboring parts with edges. (The graph in the paper by Fomin, Gaspers, Pyatkin, and Razgon, which shows that a(10) >= 105, is FCB(2, 2, 2, 2, 2).)
        n = 1: the 1-node graph;
        n = 2: the complete graph and the empty graph;
  3 <= n <= 6: the complete graph;
        n = 7: FCB(1, 2, 2, 2);
        n = 8: FCB(1, 2, 1, 2, 2);
        n = 9: FCB(1, 2, 2, 1, 3).
		

Links

• Fedor V. Fomin, Serge Gaspers, Artem V. Pyatkin, and Igor Razgon, On the minimum feedback vertex set problem: exact and enumeration algorithms, Algorithmica 52 (2008), 293-307.
• Natasha Morrison and Alex Scott, Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory Series B 126 (2017), 24-61.

Crossrefs

Cf. A000055, A005195.

Sequences of largest number of maximal induced subgraphs with a given property:

A000792 (independent sets or cliques),

this sequence (acyclic),

A342212 (bipartite),

A342213 (planar),

A342324 (chordal),

A352208 (3-colorable),

A352209 (perfect),

A352210 (2-degenerate),

A352211 (cluster graphs),

A352212 (triangle-free),

A352213 (cographs),

A352214 (claw-free),

A352215 (C_4-free),

A352216 (diamond-free).

Formula

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

1.5926... = 105^(1/10) <= lim_{n->oo} a(n)^(1/n) <= 1.8638. (Fomin, Gaspers, Pyatkin, and Razgon (2008)).