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

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

Offset: 1

Author

Jim Nastos and Clara Elliott, Jul 07 2025

Keywords

• nonn
• more

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A079473.

Programs

Sage

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")