Algorithms for enumerating and counting D2CS of some graphs

We define a D2CS of a graph G to be a set S ⊆ V (G) with diam(G[S]) ≤ 2. A D2CS arises in connection with conditional coloring and radio-k-coloring of graphs. We study the problem of counting and enumerating D2CS of a graph. We first prove the following propositions: (1) Let f (k, h) be the number of D2CS of a complete k-ary tree of height h. Then f (k, h) = k k−1 (f (k + 1, 1) − 4)(k h−1 − 1) + f (k, 1) and f (k, 1) = 2 k + k + 1. (2) A Fibonacci tree, a variant of a binary tree is defined recursively as follows: (a) Fibonacci tree of order 0 and 1 is a single node. (b) Fibonacci tree of order n (n ≥ 2) is constructed by attaching tree of order n − 2 as the leftmost child of the tree of order n − 1. Let g(n) denote the number of D2CS in a Fibonacci tree of order n. Then g(n) = 3 · 2 n−2 − (F n−1 + F n+1) + 2. (3) A binary Fibonacci tree of order n (n > 1) is a variant of a binary tree whose left subtree is of order n − 1 and right subtree of order n − 2. An order 0 Fibonacci tree has a single node, and an order 1 tree is P 2. Let h(n) denote the number of D2CS in a binary Fibonacci tree of order n. Then h(n) = 2F n + 3F n+2 − 9. (4) A binomial tree B k of order k (k ≥ 0) is an ordered tree defined recursively as: (i) B 0 is a one-vertex graph. (ii) B k consists of two copies of B k−1 such that the root of one is the leftmost child of the root of the other. Let b(k) denote the number of D2CS in a binomial tree B k. Then b(k) = k2 k + 2.