OFFSET
0,5
COMMENTS
A connected graph is called k-decomposable if it is possible to remove some edges and leave a graph with at least two connected components in which every component has at least k nodes.
Every connected graph with < 2k nodes is automatically k-indecomposable.
Necessary conditions are that a 4-indecomposable tree may not contain a path with >= 8 nodes, nor two node-disjoint paths with >= 4 nodes each.
From Brendan McKay, Feb 15 2007: (Start)
A necessary and sufficient condition seems to be that there are no two node-disjoint subtrees each of which is P_4 or K_{1,3}.
Alternatively, a tree with n vertices is k-decomposable iff, for each edge, removing that edge leaves a component with at most k-1 vertices. Finding the maximal k such that a tree is k-decomposable is easy to do in linear time. (End)
The counts of 1-indecomposable (1,0,0,0,...), 2-indecomposable (1,1,1,1,1,1,...) or 3-indecomposable (1,1,1,2,3,3,4,4,5,5,6,6,7,7,...) trees with number of nodes = 1,2,3,4,... are all trivial.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-2,-2,1,1,1,-1).
FORMULA
G.f.: f(x) / ((1-x)*(1-x^2)*(1-x^3)^2) where f(x) = 1 - x^2 - 2*x^3 + x^4 + 3*x^5 + 3*x^6 + 2*x^7 - 4*x^8 - 5*x^9 - 3*x^10 + 3*x^11 + 4*x^12 + x^13 - x^14 - x^15.
EXAMPLE
Rather than show some 4-indecomposable trees, instead we show all four 3-indecomposable trees with 7 nodes:
O-O-O-O-O....O..........O.O...O...O
....|........|..........|/.....\./.
....O....O-O-O-O-O..O-O-O-O...O-O-O
....|........|..........|....../.\.
....O........O..........O.....O...O
On the other hand, O-O-O-O-O-O-O is 3-decomposable, because removing the third edge gives O-O-O O-O-O-O, with 2 connected components each with >= 3 nodes.
PROG
(PARI) Vec((1 -x^2 -2*x^3 +x^4 +3*x^5 +3*x^6 +2*x^7 -4*x^8 -5*x^9 -3*x^10 +3*x^11 +4*x^12 +x^13 -x^14 -x^15) / ((1 -x)^4*(1 +x)*(1 +x +x^2)^2) + O(x^50)) \\ Colin Barker, May 27 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David Applegate and N. J. A. Sloane, Feb 14 2007, extended with generating function Feb 25 2007
STATUS
approved