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PlaniMatin perspectmsen walks
dcircuits
Back to minimal spanningtrees
Start connectivity Menger's theorem
Maitre representten
Vertrees numbered b in c indexingentrees invectors
makes
associated to aGmpk graph G get a hatrx
TG LTG i j0 if ii j are
notadjacent
2 if id j areadjacent
to TE
Proposition Tok jofwalks aflength k
from
i to jI 2 3
I O l e Or
fi o il
3 0 If O
I no ii ii 3
Observatory
tr TI closed walks of length 2
2 edges Edgar9degree
formula
tr CTE s cloud walks of length 3all come from D s
f G triangles in G
cyclesat length 3subgraphs
Back to
Minimalspannybeest
I
7
PnnisAlgonthyBank outfrom a gonerwhee
Start at a nlex v chosen arbitrarily
add u to our subgraph 14whichone are buildy
Eachstep add an edgeand incident
vortex of minimal
weight suchthat exactlyone ofits notices
Ceuds is inHalready
canhe until all nemangedgeshaebothends in H
H built so its alwaysconnected
maximal acyclic treeA
minimal weight fif not minimal
choose a minimal onewhoshares
first k edges wl H e e k inorder ofconsLotsof HD
Wl k maximal
H te kn has a cycleshe ti maximal
acyclic
cycle not on HE kn E H so Feo in cycle in H
natin H
in constdun eaRi i e k is stillacyclic.GreenH
w lo 3 w eatacectic
now H't en Eo is at least as light as It
and has one more edge in common
lutotomeettopio Conakily dots
Two compety conceptsofconnectivity
a robotness1fragility how many edgesneedto
be removed todisconnect
graph
redundancy howmanydistinctways can yougo
behey
denfhewtmh.es
trroomswe
DEI k G min l ofedgeswhoseremovalmakes G disconnected or Inuit
k G mill of nerfires whose removal
makes G disconnectedCartmel
Det today Gis Imd if it has a art vertex
theorem K G E K G E S G Scot minimaldynee ofa notex