The vertex leafage of chordal graphs
Discrete Applied Mathematics
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The PR-tree data structure is introduced to characterize the sets of path-tree models of path graphs. We further characterize the sets of directed path-tree models of directed path graphs with a slightly restricted form of the PR-tree called the Strong PR-tree. Additionally, via PR-trees and Strong PR-trees, we characterize path graphs and directed path graphs by their Split Decompositions. Two distinct approaches (Split Decomposition and Reduction ) are presented to construct a PR-tree that captures the path-tree models of a given graph G = (V, E) with n = |V| and m = |E|. An implementation of the split decomposition approach is presented which runs in O (nm) time. Similarly, an implementation of the reduction approach is presented which runs in O (A(n + m) nm) time (where A(s) is the inverse of Ackermann's function arising from Union-Find [40]). Also, from a PR-tree, an algorithm to construct a corresponding Strong PR-tree is given which runs in O (n + m) time. The sizes of the PR-trees and Strong PR-trees produced by these approaches are O (n + m) with respect to the given graph. Furthermore, we demonstrate that an implicit form of the PR-tree and Strong PR-tree can be represented in O (n) space.