A385697 -id:A385697 - cp's OEIS frontend

A079564 Number of unlabeled connected simple graphs on n vertices with no induced subgraphs isomorphic to a P5 or complement of a P5 (P5 = path on 5 vertices.).

Original entry on oeis.org

1, 1, 2, 6, 19, 78, 345, 1780, 10030, 62602, 429607, 3285435, 28623679, 296580989, 3897603006

Offset: 1

Jim Nastos, Jan 24 2003

J. L. Fouquet, A Decomposition for a class of (P5,P5-bar)-free graphs, Discrete Math. 121 (1993) 75-83.
V. Giakoumakis and I. Rusu, Weighted Parameters in (P5,P5-bar)-free graphs, Discrete Appl. Math. 80 (2-3) (1997) 255-261.
F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 39d0919.

Cf. A385697 (not necessarily connected).

Terms a(10)-a(15) added using tinygraph by Falk Hüffner, Jan 13 2016

Name clarified by Jim Nastos, Jul 07 2025

A385156 The number of undirected, simple, unlabeled graphs G on n vertices which are prime, not split, and do not contain a vertex of degree 1 in G or in the complement of G, and has no induced P5 in G or in the complement of G.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 22, 310, 4177

Offset: 1

Jim Nastos and Clara Elliott, Jul 22 2025

Here, "prime" means with respect to modular decomposition (see link). A P5 is a path on 5 vertices. A split graph is a graph whose vertices can be partitioned into a clique and an independent set. Related to conjectures in the referenced paper.

			a(5) = 1 is the C5.
One of the examples of a(10) = 22 is available in the links.

Maria Chudnovsky and Peter Maceli, "Simplicial Vertices in Graphs with no Induced Four-Edge Path or Four-Edge Antipath, and the H6-Conjecture," Journal of Graph Theory, vol 76, no 4, (2014).

Maria Chudnovsky and Peter Maceli, Simplicial Vertices in Graphs with no Induced Four-Edge Path or Four-Edge Antipath, and the H6-Conjecture, arXiv:1302.0404 [math.CO], 2013.
House of Graphs, One of the 22 graphs on 10 vertices.
Sean A. Irvine, Java program (github)
Wikipedia, Modular decomposition.

Cf. A385697 (when split graph condition is dropped).

a(11)-a(12) from Sean A. Irvine, Aug 17 2025

A385929 Number of simple, undirected, prime graphs G on n unlabeled vertices with no degree-1 vertex in G or its complement, as well as having no induced P5 in G or in its complement.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 9, 142

Offset: 1

Jim Nastos and Clara Elliott, Jul 12 2025

Here, "prime" means with respect to modular decomposition (see link). A P5 is a path on 5 vertices.

			The only such graph on n=5 is the C5. The only such graph on n=8 is the split graph called the 4-sun (see the House of Graphs link).

The House of Graphs, 4-sun graph
Wikipedia, Modular decomposition.

Cf. A079473, A385693, A385697.

cp's OEIS Frontend

A079564 Number of unlabeled connected simple graphs on n vertices with no induced subgraphs isomorphic to a P5 or complement of a P5 (P5 = path on 5 vertices.).

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A385156 The number of undirected, simple, unlabeled graphs G on n vertices which are prime, not split, and do not contain a vertex of degree 1 in G or in the complement of G, and has no induced P5 in G or in the complement of G.

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A385929 Number of simple, undirected, prime graphs G on n unlabeled vertices with no degree-1 vertex in G or its complement, as well as having no induced P5 in G or in its complement.

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