Power Domination Reconfiguration

1121 days ago by hogben

#################################### # This Sage worksheet is written by Chassidy Bozeman and Carolyn Reinhart # using code written by numerous other people. # The focus of this worksheet is token addition and removal (TAR) and token jumping (TJ) # reconfiguration graphs for power domination in support of the paper # Power domination reconfiguration # by # Beth Bjorkman, Chassidy Bozeman, Daniela Ferrero, Mary Flagg, # Cheryl Grood, Leslie Hogben, Bonnie Jacob, Carolyn Reinhart # # To run computations, you must first enter the function F- cells at the beginning ################################### 
       
# Functions 
       
# F-1 load main minimum rank/maximum nullity and zero forcing functions # zero forcing, PSD forcing, and minimum rank code # developed by S. Butler, L. DeLoss, J. Grout, H.T. Hall, J. LaGrange, J.C.-H. Lin,T. McKay, J. Smith, G. Tims # updated and maintained by Jephian C.-H. Lin URL='https://raw.githubusercontent.com/jephianlin/mr_JG/py2/' files=['Zq_c.pyx','Zq.py','zero_forcing_64.pyx','zero_forcing_wavefront.pyx','minrank.py', 'inertia.py'] for f in files: print("Loading %s..."%f); load(URL+f) 
        
Loading Zq_c.pyx...
Compiling
/home/sageuser/3/.sage/temp/sage.math.iastate.edu/22821/tmp_WmHttP.pyx..\
.
Loading Zq.py...
Loading zero_forcing_64.pyx...
Compiling
/home/sageuser/3/.sage/temp/sage.math.iastate.edu/22821/tmp_rENy_W.pyx..\
.
Loading zero_forcing_wavefront.pyx...
Compiling
/home/sageuser/3/.sage/temp/sage.math.iastate.edu/22821/tmp_JlJf6S.pyx..\
.
Loading minrank.py...
Loading inertia.py...
Loading Zq_c.pyx...
Compiling /home/sageuser/3/.sage/temp/sage.math.iastate.edu/22821/tmp_WmHttP.pyx...
Loading Zq.py...
Loading zero_forcing_64.pyx...
Compiling /home/sageuser/3/.sage/temp/sage.math.iastate.edu/22821/tmp_rENy_W.pyx...
Loading zero_forcing_wavefront.pyx...
Compiling /home/sageuser/3/.sage/temp/sage.math.iastate.edu/22821/tmp_JlJf6S.pyx...
Loading minrank.py...
Loading inertia.py...
# F-2 Power domination functions # by Brian Wissman # input: a graph G and a subset of its vertices V # output: true/false depending if V is a power dominating set of G def isPowerDominatingSet(G,V): N=[] for i in V: N+=G.neighbors(i) NVert=uniq(N+list(V)) A=zerosgame(G,NVert) if len(A)==G.order(): return True else: return False # input: a graph G # output: a power dominating set of G that achieves the power domination number def minPowerDominatingSet(G): V = G.vertices() for i in range(1,len(V)+1): S = subsets(V,i) for s in S: if isPowerDominatingSet(G,s): return s # input: a graph G # output: the power domination number of G def PowerDom(G): V = G.vertices() for i in range(1,len(V)+1): S = subsets(V,i) for s in S: if isPowerDominatingSet(G,s): p=len(s) return p 
       
# F-3 power dominating sets # input: a graph G and a positive integer k # output: a list of all power dominating sets G of size k def PDsets(G,k): ord=G.order() V = G.vertices() S = subsets(V,k) PDS=[] for s in S: if isPowerDominatingSet(G,s): PDS.append(s) return PDS 
       
# F-4 Token Addition and TAR and # by Chassidy Bozeman # input: A graph G and a positive integer k #output: All power dominating sets G up to size k def PDS_up_to_size_k(G,k): S=[] for i in range(1,k+1): PDS_size_i=PDsets(G,i) S=S+PDS_size_i return S #input: A graph G and a positive integer k #output: The TAR PD reconfiguration graph on power dominating set up to size k. def TAR_reconfig(G,k): S=PDS_up_to_size_k(G,k) #creates a list of all dominating sets up to size k. H=Graph() #creates empty graph H.add_vertices(S) # add power dom sets up to size k to the vertices of H # The following part determines if the size of two power dominating sets differ # by one and if one is a subset of the other. If both are true, an edge is added. for i in range(len(S)): for j in range(i+1, len(S)): if len(S[i])-len(S[j]) in {-1,1}: if set(S[i]).issubset(set(S[j])): H.add_edge(S[i],S[j]) else: if set(S[j]).issubset(set(S[i])): H.add_edge(S[i],S[j]) return H # Token Addition, Remova, Exchange TARE(Exchange=Jumping) #input: A graph G and integer k #output: The TARE PD reconfiguration graph on PDS of size up to k def TARE_reconfig(G,k): # The first part of this code is the same as that given for TAR above. S=PDS_up_to_size_k(G,k) H=Graph() H.add_vertices(S) for i in range(len(S)): for j in range(i+1,len(S)): if len(S[i])-len(S[j]) in {-1,1}: if set(S[i]).issubset(set(S[j])): H.add_edge(S[i],S[j]) else: if set(S[j]).issubset(set(S[i])): H.add_edge(S[i],S[j]) # The part below is for token exchange. If two power dominating sets have the # same size and they differ by one element, then an edge is added between them. if len(S[i])==len(S[j]): if len(set(S[i]).difference(set(S[j])))==1: H.add_edge(S[i],S[j]) return H # Token Jumping = Token Exchange def TE_reconfig(G,k): S=PDsets(G,k) #creates a list of all dominating sets of size k. H=Graph() #creates empty graph H.add_vertices(S) # add power dom sets of size k to the vertices of H # The part below is for token exchange. If two power dominating sets have the # same size and they differ by one element, then an edge is added between them. for i in range(len(S)): for j in range(i+1, len(S)): if len(S[i])==len(S[j]): if len(set(S[i]).difference(set(S[j])))==1: H.add_edge(S[i],S[j]) return H 
       
# Token Addition and Removal Reconfiguration and Complete Bipartite Graphs 
       
def TAR_reconfig_comparison(G): examples = set([]) n=G.order() H=TAR_reconfig(G,n) for g in graphs(n): K=TAR_reconfig(g,n) if H.is_isomorphic(K): if not g.is_isomorphic(G): print 'Found one' examples.add(g.graph6_string()) return examples 
       
G=graphs.CompleteBipartiteGraph(3,3) TAR_reconfig_comparison(G) 
        
set()
set()
G=graphs.CompleteBipartiteGraph(3,4) TAR_reconfig_comparison(G) 
        
set()
set()
G=graphs.CompleteBipartiteGraph(3,5) TAR_reconfig_comparison(G) 
        
set()
set()
G=graphs.CompleteBipartiteGraph(4,4) TAR_reconfig_comparison(G) 
        
set()
set()
G=graphs.CompleteBipartiteGraph(4,5) TAR_reconfig_comparison(G) 
       
# As a comparision, this program finds graphs G that are not nonisomorphic to K_2,3 # but have TAR(G) isomorphic to TAR(K_2,3): 
       
G=graphs.CompleteBipartiteGraph(2,3) TAR_reconfig_comparison(G) 
        
Found one
{'D|g'}
Found one
{'D|g'}
g=Graph("D|g") show(g) 
       
# Token Jumping Reconfiguration and Grid Graphs 
       
G5=graphs.GridGraph([5,12]) H5=TE_reconfig(G5,2) H5.is_connected() 
        
False
False
H5.connected_components() 
        
WARNING: Output truncated!
full_output.txt



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WARNING: Output truncated!
full_output.txt



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G6=graphs.GridGraph([6,12]) H6=TE_reconfig(G6,2) H6.is_connected() 
        
False
False
H6.connected_components() 
        
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G7=graphs.GridGraph([7,12]) H7=TE_reconfig(G7,2) H7.is_connected() 
        
False
False
H7.connected_components() 
        
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  ((2, 11), (3, 9)),
  ((3, 9), (4, 11)),
  ((3, 9), (5, 10)),
  ((4, 8), (5, 6)),
  ((4, 8), (5, 10)),
  ((4, 8), (6, 7)),
  ((4, 8), (6, 9))],
 [((0, 2), (2, 3)),
  ((0, 4), (2, 3)),
  ((1, 1), (2, 3)),
  ((1, 1), (3, 2)),
  ((1, 5), (2, 3)),
  ((2, 0), (3, 2)),
  ((3, 2), (4, 0)),
  ((3, 2), (5, 1)),
  ((4, 3), (5, 1)),
  ((4, 3), (5, 5)),
  ((4, 3), (6, 2)),
  ((4, 3), (6, 4))],
 [((2, 1), (4, 2)), ((4, 2), (5, 4))],
 [((1, 7), (2, 9)), ((2, 9), (4, 10))],
 [((2, 10), (4, 9)), ((4, 9), (5, 7))],
 [((1, 4), (2, 2)), ((2, 2), (4, 1))],
 [((1, 2), (2, 4))],
 [((4, 4), (5, 2))],
 [((4, 7), (5, 9))],
 [((1, 9), (2, 7))]]
[[((0, 7), (2, 8)),
  ((0, 9), (2, 8)),
  ((1, 6), (2, 8)),
  ((1, 10), (2, 8)),
  ((1, 10), (3, 9)),
  ((2, 11), (3, 9)),
  ((3, 9), (4, 11)),
  ((3, 9), (5, 10)),
  ((4, 8), (5, 6)),
  ((4, 8), (5, 10)),
  ((4, 8), (6, 7)),
  ((4, 8), (6, 9))],
 [((0, 2), (2, 3)),
  ((0, 4), (2, 3)),
  ((1, 1), (2, 3)),
  ((1, 1), (3, 2)),
  ((1, 5), (2, 3)),
  ((2, 0), (3, 2)),
  ((3, 2), (4, 0)),
  ((3, 2), (5, 1)),
  ((4, 3), (5, 1)),
  ((4, 3), (5, 5)),
  ((4, 3), (6, 2)),
  ((4, 3), (6, 4))],
 [((2, 1), (4, 2)), ((4, 2), (5, 4))],
 [((1, 7), (2, 9)), ((2, 9), (4, 10))],
 [((2, 10), (4, 9)), ((4, 9), (5, 7))],
 [((1, 4), (2, 2)), ((2, 2), (4, 1))],
 [((1, 2), (2, 4))],
 [((4, 4), (5, 2))],
 [((4, 7), (5, 9))],
 [((1, 9), (2, 7))]]
G8=graphs.GridGraph([8,12]) H8=TE_reconfig(G8,2) H8.is_connected() 
        
False
False
H8.connected_components() 
        
[[((5, 4), (6, 2))],
 [((1, 9), (2, 7))],
 [((5, 8), (6, 6))],
 [((2, 1), (4, 2))],
 [((3, 9), (5, 10))],
 [((1, 5), (2, 3))],
 [((1, 6), (2, 8))],
 [((3, 2), (5, 1))],
 [((2, 10), (4, 9))],
 [((5, 3), (6, 5))],
 [((1, 2), (2, 4))],
 [((5, 7), (6, 9))]]
[[((5, 4), (6, 2))],
 [((1, 9), (2, 7))],
 [((5, 8), (6, 6))],
 [((2, 1), (4, 2))],
 [((3, 9), (5, 10))],
 [((1, 5), (2, 3))],
 [((1, 6), (2, 8))],
 [((3, 2), (5, 1))],
 [((2, 10), (4, 9))],
 [((5, 3), (6, 5))],
 [((1, 2), (2, 4))],
 [((5, 7), (6, 9))]]