Clara Elliott - cp's OEIS frontend

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.

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.

A385693 Number of prime graphs, G, on n vertices which do not contain a degree-1 vertex in G nor in co-G.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 76, 1990, 84040, 5749698

Offset: 1

Jim Nastos and Clara Elliott, Jul 07 2025

Here, "prime" means with respect to modular decomposition (see link).

			The smallest such graph is the cycle on 5 vertices. The 6 graphs on 6 vertices are the C6, domino, X37 (as named on GraphClasses) and their three respective complements.

GraphClasses, List of Small Graphs.
Wikipedia, Modular decomposition.

Cf. A079473.

for n in range(3, 11):
    count = 0
    for g in graphs.nauty_geng(f"{n} -c -d2"):
        degrees = g.degree()
        if max(degrees) < n-2 and g.is_prime():
            count += 1
    print(f"n = {n}: {count} prime graphs")

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User: Clara Elliott

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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Author

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A385693 Number of prime graphs, G, on n vertices which do not contain a degree-1 vertex in G nor in co-G.

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