Study on Power Domination of Graphs 圖上電力支配問題的研究 研究生：莊建成 指導教授：張鎮華 Student ： Chien-Cheng Chuang Advisor ： Gerard Jennhwa Chang Department of Mathematics, National Taiwan University June

Outline Introduction Previous work and results Results in this article – Cartesian product of two cycles – Co-graphs – Trees 2

Introduction Electric power companies need to monitor the state of their power system. Let G = (V, E) represents a power system – A vertex ： an electric node (a substation bus) – An edge ： a transmission line joining two nodes One method of monitoring the system is to place phase measurement units (PMUs) in the power system. 3

Each PMU is placed on one vertex, and the observation rules of an PMU are as follows: – (1) The vertex where a PMU is placed and its incident edges are observed. – (2) The vertex that is incident to an observed edge is observed. – (3) The edge joining two observed vertices is observed. – (4) If a vertex is incident to k>1 edges and k-1 edges are observed, then the remaining edge is observed. 4

Example: 5

The system is observed if all vertices and edges are observed by a set of PMUs. G=(V,E), S is a power dominating set (PDS) if all vertices and edges are observed by S. The minimum cardinality of a power dominating set of G is called power domination number, denoted by 6

Simpler version for power domination: all vertices and edges are observed if and only if all vertices are observed. – (1) The vertex where a PMU is placed is observed. – (2) All neighbors of the vertex where the PMU is placed are observed. – (3) If a vertex has k>1 neighbors, and k-1 neighbors are observed, then the remaining neighbor is observed. 7

Example: 8

Previous work and results (1) 9

Solve power domination problem by algorithm – NP-complete: Bipartite graphs Chordal graphs Split graphs Circle graphs Planar graphs – Polynomial-time: Trees Block graphs Interval graphs Graphs of bounded treewidth ( Partial k-tree ) 10 Previous work and results (2)

Results in this article Determine the power domination numbers of Cartesian product of two cycles Find a minimum PDS for co-graphs Find a minimum PDS for trees 11

Cartesian product of two cycles 12 Some results about Cartesian product of two graphs:

13 The power domination number on grid graphs: (Dorfling-Henning, 2006)

14 Applying the method for grid graphs, we have the following theorem:

Co-graphs Disjoint union ( sum ) of two graphs Join of two graphs Definition of co-graphs – (1) – (2) 15

Proposition 1 16

Proposition 2 17

Parse tree: – the construction process of a given co-graph 18

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Trees Haynes, Hedetniemi, Hedetniemi, and Henning (2002) gave an algorithm for the power domination problem on trees. Chien (2004) gave another algorithm for trees. 20

(0,B) 21 (1,B) (2,B) (0,F) (1,B)(2,B)(1,B) (0,F)(1,F)(2,F)

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