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Power domination in block graphs

Power domination in block graphs. Guangjun Xu Liying Kang Erfang Shan Min Zhao. Outline. Introduction Preliminaries Power domination problem on trees Power domination problem in block graphs Conclusions. Introduction. Power domination

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Power domination in block graphs

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  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 powersystem. 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 • Any vertex that is incident to an observed edge is observed.

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

  6. Introduction • Observation rules • 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. Cut vertex Preliminaries • Cut vertex A cut vertex of a graph is a vertex whose deletion increases the number of components. A A

  10. Preliminaries • Block A block is a maximal biconnected subgraph of a given graph G. C E A A B B D D F F

  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. C E A B D F

  12. A b1 b2 C E B b3 A B C D D F b4 b5 E F b6 b7 Preliminaries • Cut tree BCG is a tree with vertex set and edge set

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

  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 v1,v2,…,vn so that l(vi) ≦l(vj) 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. Vr

  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 The power domination set PDS of this Tree G is {V9,V11,V12} v16 v13 v14 v15 v9 v10 v11 v12 v1 v2 v3 v4 v5 v6 v7 v8

  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. C E A B D F Power domination problems in block graphs A b1 b2 B b3 C D b4 b5 E F b6 b7

  22. Power domination problems in block graphs

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

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