Two Algorithms for Computing All Spanning Trees of a Simple, Undirected, and Connected Graph: Once Assuming a Complete Graph

Algorithm to Generate All Spanning Tree Structures of a Complete Graph

Abstract: The objective of this paper is to propose an algorithm to generate all possible structures of spanning trees of an undirected complete graph of n vertices. The process starts with a star-tree (T) of the given complete graph and then replacing the edges of T one by one to generate different possible structures like chain, branch, etc. These spanning tree structures repeat themselves as we move from lower to higher values of n. The authors have attempted to find out some generalized expressions for different structures of spanning trees for a complete graph of order n.

Graph theory with applications to engineering and computer science

Abstract: Introductory Graph Theory with ApplicationsGraph Theory with ApplicationsResearch Topics in Graph Theory and Its ApplicationsChemical Graph TheoryMathematical Foundations and Applications of Graph EntropyGraph Theory with Applications to Engineering and Computer ScienceGraphs Theory and ApplicationsQuantitative Graph TheoryApplied Graph TheoryChemical Graph TheoryA First Course in Graph TheoryGraph TheoryGraph Theory with ApplicationsGraph Theory with ApplicationsSpectra of GraphsFuzzy Graph Theory with Applications to Human TraffickingApplications of Graph TheoryChemical Applications of Graph TheoryRecent Advancements in Graph TheoryA Textbook of Graph TheoryGraph Theory and Its Engineering ApplicationsGraph Theory, Combinatorics, and ApplicationsAdvanced Graph Theory and CombinatoricsTopics in Intersection Graph TheoryGraph Theory with Applications to Engineering and Computer ScienceGraph Theory and Its Applications, Second EditionHandbook of Research on Advanced Applications of Graph Theory in Modern SocietyGraph Theory with Applications to Algorithms and Computer ScienceGraph TheoryGraph Theory with Algorithms and its ApplicationsGraph TheoryGraph Theory with ApplicationsGraph Theory ApplicationsHandbook of Graph TheoryGraph Theory and Its Applications to Problems of SocietyBasic Graph Theory with ApplicationsTen

Finding All Spanning Trees of Directed and Undirected Graphs

Abstract: An algorithm for finding all spanning trees (arborescences) of a directed graph is presented. It uses backtracking and a method for detecting bridges based on depth-first search. The time required is $O(V + E + EN)$ and the space is $O(V + E)$, where V, E, and N represent the number of vertices, edges, and spanning trees, respectively. If the graph is undirected, the time decreases to $O(V + E + VN)$, which is optimal to within a constant factor. The previously best-known algorithm for undirected graphs requires time $O(V + E + EN)$.

Algorithms for Enumerating All Spanning Trees ofUndirected and Weighted Graphs

Abstract: In this paper, we present algorithms for enumeration of spanning trees in undirected graphs, with and without weights. The algorithms use a search tree technique to construct a computation tree. The computation tree can be used to output all spanning trees by outputting only relative changes between spanning trees rather than the entire spanning trees themselves. Both the construction of the computation tree and the listing of the trees is shown to require $O(N+V+E)$ operations for the case of undirected graphs without weights. The basic algorithm is based on swapping edges in a fundamental cycle. For the case of weighted graphs (undirected), we show that the nodes of the computation tree of spanning trees can be sorted in increasing order of weight, in $O(N\log V+VE)$ time. The spanning trees themselves can be listed in $O(NV)$ time. Here $N$, $V$, and $E$ refer respectively to the number of spanning trees, vertices, and edges of the graph.

An Optimal Algorithm for Scanning All Spanning Trees of Undirected Graphs

Abstract: Let G be an undirected graph with V vertices and E edges. Many algorithms have been developed for enumerating all spanning trees in G. Most of the early algorithms use a technique called "backtracking." Recently, several algorithms using a different technique have been proposed by Kapoor and Ramesh (1992), Matsui (1993), and Shioura and Tamura (1993). They find a new spanning tree by exchanging one edge of a current one. This technique has the merit of enabling us to compress the whole output of all spanning trees by outputting only relative changes of edges. Kapoor and Ramesh first proposed an O(N + V + E)-time algorithm by adopting such a "compact" output, where N is the number of spanning trees. Another algorithm with the same time complexity was constructed by Shioura and Tamura. These are optimal in the sense of time complexity but not in terms of space complexity because they take O(VE) space. We refine Shioura and Tamura's algorithm and decrease the space complexity from O(VE) to O(V + E) while preserving the time complexity. Therefore, our algorithm is optimal in the sense of both time and space complexities.