1. Md. Abul Kashem, Chowdhury Sharif Hasan, and Anupam Bhattacharjee
An Algorithm for Solving the Minimum Vertex-Ranking Spanning Tree Problem on Series-Parallel Graphs
ICECE 2006
Md. Abul Kashem, Chowdhury Sharif Hasan, and Anupam Bhattacharjee

2. Graphs, Cycles and Trees
root
parent
nodes
child
A Graph
A cycle
A tree
Anupam Bhattacharjee, CSE, BUET

3. Spanning Trees A connected graph with no cycles is a spanning tree
In a connected cyclic graph, if we delete edges to remove cycles.
If there remains no cycle, we call it a spanning tree of the graph.
Anupam Bhattacharjee, CSE, BUET

4. SP graphs: Series Connection
Anupam Bhattacharjee, CSE, BUET

5. SP Graphs: Parallel Connection
Anupam Bhattacharjee, CSE, BUET

6. Vertex-Ranking Valid Ranking Invalid Ranking
A labeling (ranking) of the vertices of G with positive integers such that every path in G with end vertices of the same label i contains an internal vertex with label j > i.
Valid Ranking
Invalid Ranking
Anupam Bhattacharjee, CSE, BUET

7. Minimum Vertex-Ranking
A Vertex-Ranking is minimum if least number of ranks are needed to rank the graph.
A minimum vertex-ranking
A non-optimal Vertex-Ranking
Anupam Bhattacharjee, CSE, BUET

8. Minimum Vertex-Ranking Spanning Tree
The problem is to find a spanning tree of a graph whose vertex-ranking needs least number of ranks.
Input: A graph
Output: A tree with minimum vertex-ranking
Anupam Bhattacharjee, CSE, BUET

9. Binary Decomposition Tree
Anupam Bhattacharjee, CSE, BUET

10. Solution types Two types of partial solutions:
A one-tree type solution: a spanning tree is kept
A two-tree type solution: a spanning forest having exactly two components (trees) with terminal vertices in different trees is kept.
Anupam Bhattacharjee, CSE, BUET

11. Steps of the algorithm An SP graph is given:ac ab bc s
Steps of the algorithm
An SP graph is given:
Step#1: Binary decomposition tree
Step#2: Equivalence class computation for each leaf node
Anupam Bhattacharjee, CSE, BUET

12. For s-node Solution Computationab bc
Anupam Bhattacharjee, CSE, BUET

13. For p-node Solution Computationac s
DESIRED
Anupam Bhattacharjee, CSE, BUET

14. Points to note Total running time of the algorithm is O(n5 log4 n).
Some open problems still:
Develop a polylog-time parallel algorithm for solving the minimum vertex- ranking spanning tree problem on series-parallel graphs.
Develop a polynomial-time sequential algorithm for solving the minimum edge-ranking spanning tree problem on series-parallel graphs
Anupam Bhattacharjee, CSE, BUET

15. Thank you
Anupam Bhattacharjee, CSE, BUET