characterizing cohen-macaulay power edge ideals of trees
TRANSCRIPT
Characterizing Cohen-Macaulay power edge ideals oftrees
James Gossell
Clemson University
with M. Cowen, A. Hahn, W.F. Moore, S. Sather-Wagsta↵
October 21, 2020
James Gossell (Clemson) Power Edge Ideals October 21, 2020 1 / 16
Graph Theory Algebra
Electrical Engineering
1. Vertex Covers 2. Edge Ideals
3. Phasor Measurement Units (PMUs)
4. The PMU problem
6. MAIN THEOREM: Finding minimal PMU Covers
5. Power Edge Ideals
Motivation
Main Part of the Talk
Vertex Covers
Definition
Let G = (V ,E ) be a graph. A vertex cover of G is a subset V 0 ✓ V suchthat for each edge vivj in G either vi 2 V
0 or vj 2 V0. A vertex cover is
minimal if it does not properly contain another vertex cover.
Example
For the graph G = v1 v2
v4 v3
The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 3 / 16
Vertex Covers
Definition
Let G = (V ,E ) be a graph. A vertex cover of G is a subset V 0 ✓ V suchthat for each edge vivj in G either vi 2 V
0 or vj 2 V0. A vertex cover is
minimal if it does not properly contain another vertex cover.
Example
For the graph G = v1 v2
v4 v3
The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 3 / 16
Edge Ideals
Definition
Let G be a graph with vertex set V = {v1, . . . , vd}. The edge ideal of Gis the ideal IG ✓ R = A[X1, . . . ,Xd ] that is “generated by the edges of G”
IG = ({XiXj | vivj is an edge in G})R
Example
Set R = A[X1,X2,X3,X4]. The graph G = v1 v2
v4 v3
has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .
The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.We obtain the following irredundant m-irreducible decomposition:
IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .
James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16
Edge Ideals
Definition
Let G be a graph with vertex set V = {v1, . . . , vd}. The edge ideal of Gis the ideal IG ✓ R = A[X1, . . . ,Xd ] that is “generated by the edges of G”
IG = ({XiXj | vivj is an edge in G})R
Example
Set R = A[X1,X2,X3,X4]. The graph G = v1 v2
v4 v3
has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .
The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.We obtain the following irredundant m-irreducible decomposition:
IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .
James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16
Edge Ideals
Definition
Let G be a graph with vertex set V = {v1, . . . , vd}. The edge ideal of Gis the ideal IG ✓ R = A[X1, . . . ,Xd ] that is “generated by the edges of G”
IG = ({XiXj | vivj is an edge in G})R
Example
Set R = A[X1,X2,X3,X4]. The graph G = v1 v2
v4 v3
has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .
The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.
We obtain the following irredundant m-irreducible decomposition:
IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .
James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16
Edge Ideals
DefinitionThe edge ideal IG ✓ R has the following m-irreducible decompositions:
IG =\
V 0 min
PV 0
Example
Set R = A[X1,X2,X3,X4]. The graph G = v1 v2
v4 v3
has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .
The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.
We obtain the following irredundant m-irreducible decomposition:
IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .
James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16
Edge Ideals
DefinitionThe edge ideal IG ✓ R has the following m-irreducible decompositions:
IG =\
V 0 min
PV 0
Example
Set R = A[X1,X2,X3,X4]. The graph G = v1 v2
v4 v3
has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .
The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.We obtain the following irredundant m-irreducible decomposition:
IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .
James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16
Vertex Covers and Edge Ideals
Theorem (R.H. Villareal)
Let G be a tree. The following conditions are equivalent:
(i) G is unmixed. (every minimal vertex cover has the same size)
(ii) G is a suspension of a subtree G0.
(iii) IG is Cohen-Macaualay.
Example
Let
T = v1,1 v1,2 v1,3
v2,1 v2,2 v2,3
, T0 = v1,1 v1,2 v1,3
Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16
Vertex Covers and Edge Ideals
Theorem (R.H. Villareal)
Let G be a tree. The following conditions are equivalent:
(i) G is unmixed. (every minimal vertex cover has the same size)
(ii) G is a suspension of a subtree G0.
(iii) IG is Cohen-Macaualay.
Example
Let
T = v1,1 v1,2 v1,3
v2,1 v2,2 v2,3
, T0 = v1,1 v1,2 v1,3
Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16
Vertex Covers and Edge Ideals
Theorem (R.H. Villareal)
Let G be a tree. The following conditions are equivalent:
(i) G is unmixed. (every minimal vertex cover has the same size)
(ii) G is a suspension of a subtree G0.
(iii) IG is Cohen-Macaualay.
Example
Let
T = v1,1 v1,2 v1,3
v2,1 v2,2 v2,3
, T0 = v1,1 v1,2 v1,3
Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16
Vertex Covers and Edge Ideals
Theorem (R.H. Villareal)
Let G be a tree. The following conditions are equivalent:
(i) G is unmixed. (every minimal vertex cover has the same size)
(ii) G is a suspension of a subtree G0.
(iii) IG is Cohen-Macaualay.
Example
Let
T = v1,1 v1,2 v1,3
v2,1 v2,2 v2,3
, T0 = v1,1 v1,2 v1,3
Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16
Vertex Covers and Edge Ideals
Theorem (R.H. Villareal)
Let G be a tree. The following conditions are equivalent:
(i) G is unmixed. (every minimal vertex cover has the same size)
(ii) G is a suspension of a subtree G0.
(iii) IG is Cohen-Macaualay.
Example
Let
T = v1,1 v1,2 v1,3
v2,1 v2,2 v2,3
, T0 = v1,1 v1,2 v1,3
Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16
PMU Covers and Edge Ideals
DefinitionIn an electrical power system, a bus is a substation wheretransmission lines meet.
Each line connects two buses.
Example
The graphv1 v2 v3 v4
v5 v6
represents a power system with six buses and six lines.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 6 / 16
PMU Covers and Power Edge Ideals
Definition
A Phasor measurement unit (PMU) is a device placed at a bus in anelectrical power system to monitor the voltage at the bus and thecurrent in all lines connected to it. A PMU placement is a set ofbuses where PMUs are placed.
A bus in the system is observable if its voltage is known. A line is thesystem is observable if its current is known.
A PMU cover of the system is a placement of some PMUs on busesmaking the entire system observable.
Example
v1
PMU
v2 v3 v4
PMU
v5 v6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 7 / 16
PMU Covers and Power Edge Ideals
Definition
A Phasor measurement unit (PMU) is a device placed at a bus in anelectrical power system to monitor the voltage at the bus and thecurrent in all lines connected to it. A PMU placement is a set ofbuses where PMUs are placed.
A bus in the system is observable if its voltage is known. A line is thesystem is observable if its current is known.
A PMU cover of the system is a placement of some PMUs on busesmaking the entire system observable.
Example
v1PMU v2 v3 v4
PMU
v5 v6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 7 / 16
PMU Covers and Power Edge Ideals
Definition
A Phasor measurement unit (PMU) is a device placed at a bus in anelectrical power system to monitor the voltage at the bus and thecurrent in all lines connected to it. A PMU placement is a set ofbuses where PMUs are placed.
A bus in the system is observable if its voltage is known. A line is thesystem is observable if its current is known.
A PMU cover of the system is a placement of some PMUs on busesmaking the entire system observable.
Example
v1PMU v2 v3 v4
PMU
v5 v6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 7 / 16
PMU Covers and Power Edge Ideals
Definition
A Phasor measurement unit (PMU) is a device placed at a bus in anelectrical power system to monitor the voltage at the bus and thecurrent in all lines connected to it. A PMU placement is a set ofbuses where PMUs are placed.
A bus in the system is observable if its voltage is known. A line is thesystem is observable if its current is known.
A PMU cover of the system is a placement of some PMUs on busesmaking the entire system observable.
Example
v1PMU v2 v3 v4
PMU
v5 v6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 7 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
v
PMU
1
obs.
v
obs.
2
obs.
obs.
v
obs.
3
obs.
obs.
v
PMU
4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
v
PMU
1
obs.
v
obs.
2
obs.
obs.
v
obs.
3
obs.
obs.
v
PMU
4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.
v
obs.
2
obs.
obs.
v
obs.
3
obs.
obs.
v
PMU
4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.v
obs.
2
obs.
obs.
v
obs.
3
obs.
obs.
v
PMU
4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.vobs.2
obs.
obs.
v
obs.
3
obs.
obs.
v
PMU
4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.vobs.2
obs.
obs.
v
obs.
3
obs.
obs.
vPMU4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.vobs.2
obs.
obs.
v
obs.
3
obs.
obs.vPMU4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.vobs.2
obs.
obs.
vobs.3
obs.
obs.vPMU4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.vobs.2
obs.
obs.vobs.3
obs.
obs.vPMU4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.vobs.2
obs.
obs.vobs.3
obs.
obs.vPMU4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.vobs.2
obs.
obs.vobs.3
obs.
obs.vPMU4
vobs.5
obs.
vobs.6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to an observable vertex v are
observable except for one, then all of the lines incident to v are
observable.
Example
vPMU1
obs.vobs.2
obs.
obs.vobs.3
obs.
obs.vPMU4
vobs.5
obs.vobs.6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to a vertex v are observable except
for one, then all of the lines incident to v are observable.
Example
v
obs.
1
obs.
vPMU2
obs.
obs.
v
obs.
3
obs.
obs.
v
obs.
4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to a vertex v are observable except
for one, then all of the lines incident to v are observable.
Example
v
obs.
1obs.
vPMU2
obs.
obs.v
obs.
3
obs.
obs.
v
obs.
4
v
obs.
5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to a vertex v are observable except
for one, then all of the lines incident to v are observable.
Example
vobs.1
obs.vPMU2
obs.
obs.vobs.3
obs.
obs.
v
obs.
4
vobs.5
obs.
v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to a vertex v are observable except
for one, then all of the lines incident to v are observable.
Example
vobs.1
obs.vPMU2
obs.
obs.vobs.3
obs.
obs.
v
obs.
4
vobs.5
obs.v
obs.
6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to a vertex v are observable except
for one, then all of the lines incident to v are observable.
Example
vobs.1
obs.vPMU2
obs.
obs.vobs.3
obs.
obs.
v
obs.
4
vobs.5
obs.vobs.6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to a vertex v are observable except
for one, then all of the lines incident to v are observable.
Example
vobs.1
obs.vPMU2
obs.
obs.vobs.3
obs.
obs.
v
obs.
4
vobs.5
obs.vobs.6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to a vertex v are observable except
for one, then all of the lines incident to v are observable.
Example
vobs.1
obs.vPMU2
obs.
obs.vobs.3
obs.
obs.v
obs.
4
vobs.5
obs.vobs.6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16
PMU Covers and Power Edge Ideals
Rule1 Place a PMU at vertex v . Then the vertex v is observable and every
line incident to v is observable.
2 (Ohm) A bus incident to an observable line is observable.
3 (Ohm) A line incident to two observable buses is observable.
4 (Kircho↵) If all the lines incident to a vertex v are observable except
for one, then all of the lines incident to v are observable.
Example
vobs.1
obs.vPMU2
obs.
obs.vobs.3
obs.
obs.vobs.4
vobs.5
obs.vobs.6
James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16
PMU Covers and Power Edge Ideals
Definition
Let G be a graph with vertex set V = {v1, . . . , vd} and setR = A[X1, . . . ,Xd ]. The power edge ideal of G is
IPG =
\
V 0
PV 0
where the intersection is taken over all PMU covers of G .
Example
v1 v2 v3 v4
v5 v6
IPG = hX1,X4i \ hX2i \ hX3i \ hX5i \ hX6i = hX1X2X3X5X6,X2X3X4X5X6i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 10 / 16
PMU Covers and Power Edge Ideals
Definition
Let G be a graph with vertex set V = {v1, . . . , vd} and setR = A[X1, . . . ,Xd ]. The power edge ideal of G is
IPG =
\
V 0
PV 0
where the intersection is taken over all PMU covers of G .
Example
v1 v2 v3 v4
v5 v6
IPG = hX1,X4i \ hX2i \ hX3i \ hX5i \ hX6i = hX1X2X3X5X6,X2X3X4X5X6i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 10 / 16
The PMU Placement Problem
The PMU Placement ProblemDetect any and all outages in the system and minimize cost.I.e., find the smallest number of PMUs needed to monitor the entiresystem.
A Problem with the PMU Placement Problem
The PMU Placement Problem is NP-complete (HHHH).
GoalStudy the problem algebraically and find some interesting examples ofideals along the way. Specifically, we will:
Characterize the trees T such that IPT is unmixed, i.e., such that allminimal PMU covers have the same size.
Characterize the trees T such that IPT is Cohen-Macaulay.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 11 / 16
The PMU Placement Problem
The PMU Placement ProblemDetect any and all outages in the system and minimize cost.I.e., find the smallest number of PMUs needed to monitor the entiresystem.
A Problem with the PMU Placement Problem
The PMU Placement Problem is NP-complete (HHHH).
GoalStudy the problem algebraically and find some interesting examples ofideals along the way. Specifically, we will:
Characterize the trees T such that IPT is unmixed, i.e., such that allminimal PMU covers have the same size.
Characterize the trees T such that IPT is Cohen-Macaulay.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 11 / 16
The PMU Placement Problem
The PMU Placement ProblemDetect any and all outages in the system and minimize cost.I.e., find the smallest number of PMUs needed to monitor the entiresystem.
A Problem with the PMU Placement Problem
The PMU Placement Problem is NP-complete (HHHH).
GoalStudy the problem algebraically and find some interesting examples ofideals along the way. Specifically, we will:
Characterize the trees T such that IPT is unmixed, i.e., such that allminimal PMU covers have the same size.
Characterize the trees T such that IPT is Cohen-Macaulay.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 11 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:
(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
Example
v1
PMU obs.
v2
obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs. obs.
v7
obs. obs.
v8
obs.
obs.
obs.
v9
PMU
v10
obs. obs.
v11
PMU obs.
v12
obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.
(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
Example
v1
PMU obs.
v2
obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs. obs.
v7
obs. obs.
v8
obs.
obs.
obs.
v9
PMU
v10
obs. obs.
v11
PMU obs.
v12
obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.
(iii) G is obtained by “nicely linking” a sequence of paths.-PMU Covers: Choose one vertex from each path
Example
v1
PMU obs.
v2
obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs. obs.
v7
obs. obs.
v8
obs.
obs.
obs.
v9
PMU
v10
obs. obs.
v11
PMU obs.
v12
obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU
obs.
v2
obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs. obs.
v7
obs. obs.
v8
obs.
obs.
obs.
v9PMU
v10
obs. obs.
v11PMU
obs.
v12
obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2
obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs. obs.
v7
obs. obs.
v8
obs.
obs.
obs.v9
PMU
v10
obs.
obs.v11
PMU obs.v12
obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs. obs.
v7
obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs. obs.
v7
obs.
obs.v8
obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs. obs.
v7obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6
obs.
obs.v7
obs. obs.v8
obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6obs. obs.
v7obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.
v3
obs. obs.
v4
obs.
v5
obs. obs.
v6obs. obs.
v7obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.v3
obs. obs.
v4
obs.
v5
obs.
obs.v6
obs. obs.v7
obs. obs.v8
obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.v3
obs.
obs.
v4
obs.
v5obs. obs.
v6obs. obs.
v7obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.v3
obs. obs.v4
obs.
v5obs. obs.
v6obs. obs.
v7obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-PMU Covers: Choose one vertex from each path
Example
v1PMU obs.
v2obs.
obs.
obs.v3
obs. obs.v4
obs.
v5obs. obs.
v6obs. obs.
v7obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-Generators of IPG : The product of the vertices in the paths.
Example
v1PMU obs.
v2obs.
obs.
obs.v3
obs. obs.v4
obs.
v5obs. obs.
v6obs. obs.
v7obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Characterizing Unmixed Trees
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.
-Generators of IPG : The product of the vertices in the paths.
Example
v1PMU obs.
v2obs.
obs.
obs.v3
obs. obs.v4
obs.
v5obs. obs.
v6obs. obs.
v7obs. obs.
v8obs.
obs.
obs.v9
PMU
v10obs. obs.
v11PMU obs.
v12obs.
IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i
James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16
Alegraic Implications
Summary
For any ideal I , we have
I is a CI =) I is Gor. =) I is CM =) I is unmixed
For any tree T , we have
IPT is a CI () I
PT is Gor. () I
PT is CM () I
PT is unmixed
ClaimIn general, for abitrary graphs G ,
IPG is a CI 6 I(=
P
G is Gor. 6 I(=P
G is CM 6 I(=P
G is unmixed
James Gossell (Clemson) Power Edge Ideals October 21, 2020 13 / 16
Alegraic Implications
Summary
For any ideal I , we have
I is a CI =) I is Gor. =) I is CM =) I is unmixed
For any tree T , we have
IPT is a CI () I
PT is Gor. () I
PT is CM () I
PT is unmixed
ClaimIn general, for abitrary graphs G ,
IPG is a CI 6 I(=
P
G is Gor. 6 I(=P
G is CM 6 I(=P
G is unmixed
James Gossell (Clemson) Power Edge Ideals October 21, 2020 13 / 16
Alegraic Implications
Summary
For any ideal I , we have
I is a CI =) I is Gor. =) I is CM =) I is unmixed
For any tree T , we have
IPT is a CI () I
PT is Gor. () I
PT is CM () I
PT is unmixed
ClaimIn general, for abitrary graphs G ,
IPG is a CI 6 I(=
P
G is Gor. 6 I(=P
G is CM 6 I(=P
G is unmixed
James Gossell (Clemson) Power Edge Ideals October 21, 2020 13 / 16
Summery
Theorem (Villareal)
Let G be a tree. The following conditions are equivalent:
(i) G is unmixed with respect to vertex covers.
(ii) G is a suspension of a subtree G0.
(iii) Every vertex of deg. � 2 is adjacent to exactly 1 vertex of deg. 1.(iv) IG is Cohen-Macaualay.
Theorem (CGHMS)
Let G be a tree. The following conditions are equivalent:
(i) G is unmixed with respect to PMU covers.
(ii) G is obtained by “nicely linking” a sequence of paths.
(iii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iv) I
PG is Cohen-Macaulay.
(v) IPG is Gorenstein and Complete Intersection.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 14 / 16
Counterexamples
Claim
IPG is a CI 6(= I
PG is Gor. 6(= I
PG is CM 6(= I
PG is unmixed
James Gossell (Clemson) Power Edge Ideals October 21, 2020 15 / 16
Counterexamples
Claim
IPG is a CI 6
?
(= IPG is Gor. 6(= I
PG is CM 6(= I
PG is unmixed
Example (IPG can be Gorenstein but not a Complete Intersection)
v1
v2 v3
v4 v7
v5 v8
v6 v9
v10 v11 v15 v13
v12 v14James Gossell (Clemson) Power Edge Ideals October 21, 2020 15 / 16
Counterexamples
Claim
IPG is a CI 6(= I
PG is Gor. 6
?
(= IPG is CM 6(= I
PG is unmixed
Example (IPG can be Cohen-Macaulay but not Gorenstein)
v1 v2 v3
v6 v5 v4
James Gossell (Clemson) Power Edge Ideals October 21, 2020 15 / 16
Counterexamples
Claim
IPG is a CI 6(= I
PG is Gor. 6(= I
PG is CM 6
?
(= IPG is unmixed
Example (IPG can be Unmixed but not Cohen-Macaulay)
v1 v2 v3 v4 v5
v6 v7 v8 v9 v10
James Gossell (Clemson) Power Edge Ideals October 21, 2020 15 / 16
References
Villareal, Rafael H., 1990
Cohen-macaulay graphs
Manuscripta Mathematica 66(1), 277-293.
Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi, Michael A.
Henning 2002
Domination in Graphs Applied to Electric Power Networks
SIAM J. Discrete Math 15(4), 519-529.
W. Frank Moore, Mark Rogers, Sean Sather-Wagsta↵ 2017
Monomial Ideals and Their Decompositions.
James Gossell (Clemson) Power Edge Ideals October 21, 2020 16 / 16