Counting Paths and Packings in Halves

Abstract

Abstract. It is shown that one can count k-edge paths in an n-vertex graph and m-set k-packings on an n-element universe, respectively, in time`n k/2´a nd`n mk/2´, up to a factor polynomial in n, k, and m; in polynomial space, the bounds hold if multiplied by 3 k/2 or 5 mk/2 , respectively. These are implications of a more general result: given two set families on an n-element universe, one can count the disjoint pairs of sets in the Cartesian product of the two families with O(n ) basic operations, where is the number of members in the two families and their subsets.

On the number of paths of length 5 in a graph

“…However, in the cases with more than one figure (Cases 5,6,7,8,9,10,11,14,15,17,18,19,22 and 23), N, M and F are based on the first graph of the respective figures and P 1 , P 2 ,... denote the number of subgraphs of G which do not have the same configuration as the first graph but are counted in M. It is clear that F is equal to N × (M− P 1 −P 2 −...). To find N in each case, we have to include in any walk, all the edges and the vertices of the corresponding subgraphs at least once.…”

On the number of paths of length 6 in a graph

“…In 1996, Eric Bax and Joel Franklin [7], gave an algorithm to count paths and cycles of a given length in a directed graph. In [6,8,9,10,12,13,15], we have also some bounds to estimate the total time complexity for finding or counting paths and cycles in a graph. In the previous works there is no formula to count the exact number of paths of a specific size in a graph.…”

On the Number of Paths of Lengths 3 and 4 in a Graph

“…In [4], [6], [7], [8], [10], [11] and [13], we have also some bounds to estimate the total time complexity for finding or counting paths and cycles in a graph. In the previous works there is no formula to count the exact number of paths of an specific size in a graph.…”