Algorithms for enumerating multiple leaf-distance granular regular α-subtree of unicyclic and edge-disjoint bicyclic graphs

A multiple leaf-distance granular regular α-tree (abbreviated as LDR α-tree for short) is a tree (with at least α+1 vertices) where any two leaves are at some distance divisible by α. A connected graph's subtree which is additionally an LDR α-tree is known as an LDR α-subtree. Obviously, α=1 and 2, correspond to the general subtrees (excluding the single vertex subtrees) and the BC-subtrees (the distance between any two leaves of the subtree is even), respectively. With generating functions and structure decomposition, in this paper, we propose algorithms for enumerating an auxiliary subtree α_τ(v)-subtree (τ=0,1,…,α−1) containing a fixed vertex, and various LDR α-subtrees of unicyclic graphs, respectively. Basing on these algorithms, we further present algorithms for enumerating various LDR α-subtrees of edge-disjoint bicyclic graphs.