1. Power domination in block graphs Guangjun Xu Liying Kang Erfang Shan Min Zhao

2. Outline Introduction Preliminaries Power domination problem on trees Power domination problem in block graphs Conclusions

3. Introduction Power domination Let G = (V,E) be a graph representing an electric power system. A PMU measures the state variable (voltage and phase angle) for the vertex at which it is placed and its incident edges and their endvertices. (These vertices and edges are said to be observed.) We will introduce the observation rules [1] in the following. Power domination problem is to observe all the electric power system.

4. Introduction Observation rules 1. Any vertex that is incident to an observed edge is observed.

5. Introduction Observation rules 2. Any edge joining two observed vertices is observed.

6. Introduction Observation rules 3. If a vertex is incident to a total of k >1 edges and k-1 of these edges are observed, then all k of these edges are observed.

7. Introduction

8. Preliminaries Cut vertex Block Block graphs Cut tree

9. Preliminaries Cut vertex A cut vertex of a graph is a vertex whose deletion increases the number of components. AA Cut vertex

10. Preliminaries Block A block is a maximal biconnected subgraph of a given graph G. AB C DF E AB DF

11. Preliminaries Block graphs G is a block graph If all blocks are complete graphs and the intersection of two blocks is either empty or a cut vertex. AB C DF E

12. Preliminaries Cut tree BC G is a tree with vertex set and edge set A b1b1 B CD FE b2b2 b3b3 b4b4 b5b5 b7b7 b6b6 AB C DF E

13. Some observed properties about PDS: 1. 2. 3. a. b. * A vertex adjacent to two or more leaves is called a strong support vertex. Preliminaries

14. Power domination problem on trees A linear time algorithm in tree  Input: A tree G on n ≧ 2 vertices rooted at a vertex of maximum degree with the vertices labeled v 1,v 2,…,v n so that l(v i ) ≦ l(v j ) for i>j.  Output: Power domination set: S A partition of V(G) into |S| subsets:

15. Power domination problem on trees A linear time algorithm in tree  Step 1: Check if G is a spider. If true then quit. VrVr

16. Power domination problem on trees A linear time algorithm in tree  Step 2: If v is a leaf or a non-dominated vertex that is not strong support vertex.

17. Power domination problem on trees A linear time algorithm in tree  Step 3: If v is a strong support vertex.

18. Power domination problem on trees v1v1 v2v2 v9v9 v 13 v 16 v 15 v 12 v6v6 v7v7 v8v8 v 14 v 10 v 11 The power domination set PDS of this Tree G is {V 9,V 11,V 12 } v3v3 v4v4 v5v5

19. Power domination problems in block graphs Deal with those vertices that may be dominated by observation rules 2 and 3.

20. Power domination problems in block graphs Decide the color of every vertices.

21. Power domination problems in block graphs A b1b1 B CD FE b2b2 b3b3 b4b4 b5b5 b7b7 b6b6 AB C DF E

23. Conclusion They proposed a linear-time algorithm for power domination problem in block graphs.