Counting spanning trees in graphs using modular decomposition

Question

1. What contributions have the authors mentioned in the paper "Counting spanning trees in graphs using modular decomposition" ?

2. What is the importance of a spanning tree in graph theory?

3. What is the function for determining the determinant of a graph?

4. What is the determinant of matrix M?

Accepted Answer

In this paper the authors present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, their algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node ; then, the number of spanning trees of G is computed as the product of a collection of values which are associated with the vertices of G and are updated during the contraction process.. Therefore the authors give the first linear-time algorithm for the counting problem in the considered graph

Accepted Answer

In particular, counting spanning trees is an essential step in many methods for computing, bounding, and approximating network reliability [8]; in a network modeled by a graph, intercommunication between all nodes of the network implies that the graph must contain a spanning tree and, thus, maximizing the number of spanning trees is a way of maximizing reliability.

Accepted Answer

The elimination is applied on a contractible node t and is done by means of two functions, namely, Handle-Basic, in the case where Gt belongs to one of the graph classes of the family of basic graphs, and Handle-NonBasic, otherwise.

Accepted Answer

In order to compute the determinant of matrix M the authors zero the off-diagonal elements formed by the p× p submatrix of the first p rows and p columns.

Accepted Answer

In the end, the number of spanning trees of G is obtained as the product of n numbers, where n is the number of vertices ofG; this multiplication takes O(n) time under the uniform-cost criterion [1,23].

Accepted Answer

The Kirchhoff matrix tree theorem (Theorem 1) implies that the number of spanning trees of a graph G on n vertices is equal to any of the cofactors of the Kirchhoff matrix K, and thus, to the determinant of the (n − 1) × (n − 1) submatrix M of K formed by the first n− 1 rows and the first n− 1 columns.

Accepted Answer

For an n× n matrix M , the (n− 1)-st order minor μij is the determinant of the (n−1)×(n−1) matrix obtained from M after having deleted row i and column j.

Accepted Answer

Note that the parameter −1 of the functions for the elimination of an edgeless graph, a thin spider, a prime tree, or a chordless cycle is to signal that the input graph is edgeless, a thin spider, a prime tree, or a chordless cycle and not their complements.

Accepted Answer

Since computing the number of spanning trees of a graph on a fixed number of vertices takes constant time, Theorem 2 implies that the non-basic cost of a (q, q − 4)-graph G on n vertices is φ(G) = O(n), for every fixed q ≥ 4.

Accepted Answer

Their algorithm is easy to implement; its correctness is established by means of the Kirchhoff matrix tree theorem along with standard techniques from linear algebra and matrix theory.

Accepted Answer

The authors associate each of the n− 1 vertices with an s-value which is equal to the vertex’s degree in G, i.e, s(vi) = d(vi), where d(vi), 1 ≤ i ≤ n − 1, is the degree of vertex vi in graph G.

Accepted Answer

Other interesting problems involve counting other structures as well, e.g., the number of perfect matchings, Hamiltonian cycles and Euler cycles.

Accepted Answer

As described in the previous section, in order to compute the determinant of matrix M the authors zero the off-diagonal elements formed by the p × p submatrix of the first p rows and p columns.

Accepted Answer

Note that, the form of this submatrix depends on the structure of the graph Gt (i.e., on the type of the internal node t of T (G)).

Accepted Answer

Gt is isomorphic to the graph induced by a subset of Mt consisting of a single vertex from each maximal submodule of Mt in T (G).

Counting spanning trees in graphs using modular decomposition

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Most frequently asked questions

1. What contributions have the authors mentioned in the paper "Counting spanning trees in graphs using modular decomposition" ?

2. What is the importance of a spanning tree in graph theory?

3. What is the function for determining the determinant of a graph?

4. What is the determinant of matrix M?

5. How many spanning trees of G are obtained?

6. What is the Kirchhoff matrix tree theorem?

7. What is the determinant of the (n1)(n1) matrix?

8. What is the function for the elimination of an edgeless graph?

9. What is the non-basic cost of a tree graph?

10. How is the modular decomposition algorithm established?

11. What is the degree of vertex vi in graph G?

12. What other interesting problems involve counting other structures?

13. What is the function for determining the determinant of matrix M?

14. What is the form of the submatrix?

15. What is the graph induced by a subset of Mt?

Figures

Citations

Algebraic Graph Theory: The multiplicative expansion

References

The Design and Analysis of Computer Algorithms

Algebraic Graph Theory

Algebraic graph theory

Matrix multiplication via arithmetic progressions

Matrix multiplication via arithmetic progressions

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