A Linear Time Algorithm for Computing Longest Paths in Cactus Graphs

Abstract

We propose an algorithm that computes the length of a longest path in a cactus graph. Our algorithm can easily be modified to output a longest path as well or to solve the problem on cacti with edge or vertex weights. The algorithm works on rooted cacti and assigns to each vertex a two-number label, the first number being the desired parameter of the subcactus rooted at that vertex. The algorithm applies the divide-and-conquer approach and computes the label of each vertex from the labels of its children. The time complexity of our algorithm is linear in the number of vertices, thus improving the previously best quadratic time algorithm.

Programming and Testing a Two-Tree Algorithm

“…One line will contain the endpoint a and the other the endpoint b of an existing edge, (a, b), on the two-tree. Both line segments will contain w as the other endpoint (Figures 1 and 2) [2].…”

“…The Algorithm. All non-trivial two-trees can be viewed as a graph of triangular faces attached along their edges, where each face is identified by it's three vertices (x, y, z) (Figure 2) [2]. Considering a two-tree in this fashion, all edges can be classified as either periphery or interior.…”

“…Considering a two-tree in this fashion, all edges can be classified as either periphery or interior. A periphery edge is an edge that belongs to a single face and an interior edge is common to multiple faces [2]. From the two-tree in Figure 2, it is apparent that edge (w, v) is peripheral while edge (v, u) is interior.…”

“…. ) and in each subgraph, whose root edge is also (u, v), the root edge is peripheral [2] (Figures 3 and 4). If the root edge of G is peripheral, G⊖e, we will find a single At this point trying to break down any of the subgraphs by this process of splitting on edge G i ⊖e would be futile.…”

“…4. The subgraphs resulting from (G − (w, v)) ⊖ u e, is associated with a label λ(e), which systematically keeps track of the tree's longest path and the lengths of the longest paths with starting or ending vertices at the roots of the tree [2]. This later information is needed for finding the longest path of a tree G from the labels of its subtrees G 1 , G 2 , G 3 , .…”