Recovering Conductances of Resistor Networks in a Punctured Disk

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1 Recovering Conductnces of Resistor Networks in Punctured Disk Yuli Alexndr Brin Burks Ptrici Commins Novemer 27, 2018 Astrct The response mtrix of resistor network is the liner mp from the potentil t the oundry vertices to the net current t the oundry vertices. For circulr plnr resistor networks, Curtis, Ingermn, nd Morrow hve given necessry nd sufficient condition for recovering the conductnce of ech edge in the network uniquely from the response mtrix using locl moves nd medil grphs. We generlize their results for resistor networks on punctured disk. First we discuss dditionl locl moves tht occur in our setting, prove severl results out medil grphs of resistor networks on punctured disk, nd define the notion of z-sequences for such grphs. We then define certin circulr plnr grphs tht re electriclly equivlent to stndrd grphs nd turn them into networks on punctured disk y dding oundry vertex in the middle. We prove such networks re recoverle nd re le to generlize this result to much roder fmily of networks. A necessry condition for recoverility is lso introduced. 1 Introduction In this pper, we study resistor networks, electricl networks mde up of only resistors. The electricl properties of these networks is chrcterized y the response mtrix. The response mtrix gives the liner mp from potentil ssignment t ech oundry vertex to the net current flow t ech vertex. A common question in the study of electricl networks is when we cn uniquely recover the conductnces of resistor network given the response mtrix. This is clled the Dirichlet-to-Neumnn prolem. This prolem hs een extensively studied for clss of networks clled circulr plnr resistor networks. Curtis, Moores, nd Morrow [2] developed n lgorithm to recover the conductnces of prticulr fmily of these grphs clled stndrd grphs. Curtis, Ingermn, nd Morrow [1] nd De Verdiere, Gitler, nd Vertign [3] were le to use this lgorithm to otin results for ll circulr plnr resistor networks. They considered the property of criticlity, condition regrding the connections of network. Building on the previous lgorithm nd using medil grphs, they proved tht the networks where we uniquely recover conductnces from the response mtrix re exctly the criticl networks. They were lso le to use medil grphs to show complete ctlogue of locl moves tht preserve the response mtrix of circulr plnr resistor network nd use these locl moves to clssify criticl networks. Moreover, they were le to identify ll response mtrices of resistor networks with positivity criteri for minors of the mtrix. Kenyon nd Wilson [5, 6] studied the comintorics of the response mtrix for resistor networks. In prticulr, they were le to relize the minors of the response mtrix for circulr plnr resistor Wesleyn University, CT University of Cliforni, Berkeley, CA Crleton College, MN 1

2 network s sum over groves in the network. This explins the positivity criteri for response mtrices of circulr plnr resistor networks found in erlier work. However, very little is known for more complicted networks. In [7], Lm nd Pylyvskyy studied the Dirichlet-to-Neumnn prolem for electricl networks on cylinder. They gve conjecturl solution for generl cylindricl electricl networks, nd showed it held for specil clss of these networks known s purely cylyndricl networks. In this pper, we introduce resistor networks in punctured disk (rnpds). These re networks emedded in disk where exctly one oundry vertex is plced inside the disk. We first discuss new locl moves tht rise for these networks. We conjecture tht the locl moves stted in the pper re the only moves tht preserve electricl equivlence. Next we define medil grphs for rnpds nd prove severl properties of the medil grph for irreducile rnpds. Medil grphs llow us to define spider grphs, fmliy of rnpds closely relted to the stndrd grphs. Using spider grphs, we find severl conditionl locl moves tht cn e pplied to rnpds under certin conditions without chnging the response mtrix. Turning to the question of recoverility, we find clss of recoverle rnpds. We lso give necessry condition for recoverility or rnpds. 2 Bckground nd Definitions For more ckground on electricl networks, see [1], [3], nd [4]. Defnition 1. A resistor network is grph (V, E) with specified set B V of oundry vertices nd rel nonnegtive conductnce c e, for ech e E. The remining vertices, I = V \ B, re clled internl vertices. We will sy grph to refer to resistor network without conductnces. In this pper, we ll ssume ll resistor networks re connected grphs. Our convention throughout this pper will e to color oundry vertices white nd internl vertices lck. Defnition 2. A circulr plnr resistor network (cprn) is resistor network tht cn e emedded in disk so tht it is plnr nd ll the oundry vertices re on the oundry of the disk. A resistor network in punctured disk (rnpd) is resistor network tht cn e emedded in disk so tht it is plnr nd ll oundry vertices ut one re on the oundry of the disk. See Figure Figure 1: A cprn (left) nd rnpd (right) Defnition 3. A non-trivl rnpd is n rnpd tht cnnot e drwn s cprn. Note tht in non-trivil rnpd, the interior oundry vertex of must e surrounded y polygon. Otherwise, we re le to redrw the network so tht the interior oundry vertex is on the oundry of the disk. Defnition 4. A potentil function for resistor network (V, E) is function f : V R 0. An edge (v 1, v 2 ) hs voltge given y the difference in potentil t v 1 nd v 2. We wnt our resistor networks to model electricl circuits. This mens they need to stisfy Ohm s Lw nd Kirchhoff s Current Lw. 2

3 Theorem 1 (Ohm s Lw). If v e is the voltge cross edge e, then v e c e = i e, where i e is the current cross edge e. By Ohm s Lw, potentil function induces current cross ech edge. Note tht current lwys flows from the vertex with the higher potentil to the vertex with the lower potentil. Theorem 2 (Kirchhoff s Current Lw). For ech node in n electricl circuit, the sum of the currents entering the node must e the sme s the sum of the currents leving the node. The nlogue of Kirchhoff s Current Lw for resistor networks is tht for ech internl vertex in resistor network, the sum of the currents entering the vertex must e the sme s the sum of the currents leving the vertex. If we use Ohm s Lw to clculte current, only certin potentil functions stisfy Kirchhoff s current Lw. Theorem 3 ([1]). Given function φ : B R 0 for resistor network, there is unique choice of potentil function f where f B = φ nd the currents induced y f from Ohm s Lw stisfy Kirchhoff s Current Lw. Defnition 5. Let Γ = (V, E) e resistor network with m vertices, n of which re oundry vertices. Index ll oundry vertices v B s {1,, n} nd ll internl vertices v I s {n + 1,, m}. Then, we define the Kirchhoff mtrix of Γ s follows: c e i j e=(i,j) K i,j = i = j e=(i,k) k [n] Note tht the Kirchhoff mtrix cn e represented s: A K = B T where A is symmetric n n mtrix, B is n n (m n), nd C is symmetric (m n) (m n) mtrix. Defnition 6. The response mtrix, Λ(Γ) of resistor network Γ is the Schur complement A BC 1 B T of the Kirchhoff mtrix. We cn represent potentil ssignment to the oundry vertices s vector u R n 0. Λ(Γ) gives the liner mp from such n ssignment to the vector of currents flowing into ech oundry vertex (where negtive vlue represents current flowing out of the vertex). Exmple 1. Consider the following cprn: c e B C c 3 1 d 3

4 We see tht the vertex set is V = {,, c, d}. The set of oundry vertices is B = {, c} nd the set of internl vertices is I = {, d}. Thus, in order to compute the Kirchhoff mtrix, we lel nd c y 1 nd 2, respectively, nd nd d y 3 nd 4, respectively. Then, the Kirchhoff mtrix is: K = Now, s descried ove, one cn compute the response mtrix: T = Defnition 7. Grphs Γ 1 nd Γ 2 re electriclly equivlent, if, for every ssignment of conductnces for Γ 1, there exists n ssignment of conductnces to Γ 2 such tht the resulting electricl networks hve the sme response mtrix, nd vice vers. We re interested in three properties of networks tht hve een shown to e the sme for the cprn cse: irreduciility, recoverility, nd criticlity. Defnition 8. We cll grph reducile if it is electriclly equivlent to grph with fewer edges, nd irreducile otherwise. Note tht network my hve the sme response mtrix s network with fewer edges while still eing irreducile. Defnition 9. A resistor network is recoverle if given its response mtrix nd nd grph, we re le to uniquely determine its conductnces. Defnition 10. A connection of resistor network Γ = (V, E) is tuple (P, Q) s.t. P = {p 1,, p k } nd Q = {q 1,, q k } re susets of B nd there exists k disjoint pths tht do not pss through oundry vertices connecting p i to q π(i), i nd for some π S k. For cprn, we require tht p 1,, p k, q k,, q 1 occur in clockwise order round the disk s oundry. For n rnpd, we hve the sme requirement, with the exception tht the interior oundry vertex cn pper in either Q or P, t ny index. Remrk 1. If Γ is cprn nd (P, Q) is connection, then the only possile π S k connecting P nd Q is the identity permuttion. Otherwise, the pths will not e disjoint. Defnition 11. A cprn is criticl if the removl of ny edge in the network reks connection. Curtis, Ingermn, nd Morrow [1] proved tht these three definitions re equivlent in the cprn cse. In this pper, we investigte to wht extent this equivlence holds in the rnpd cse. 4

5 3 Locl Moves There re five locl trnsformtions on cprns tht do not chnge the response mtrix [3]. We list these elow. Note tht the networks we drw elow re components of possily lrger network. The grey vertices indicte where the components in the pictures my e connected to the rest of the grph. Tht is, they re vertices tht my e either oundry vertices or internl vertices nd my hve other edges incident to them. In the cses where the rrow goes only in one direction, we my still pply the trnsformtion in the opposite direction, ut there is not unique wy to do so. Loop removl Pendnt removl Series trnsformtion + Prllel trnsformtion + Y- move c A B = = c = AB + AC + BC A AB + AC + BC B AB + AC + BC C C c A = + + c c B = + + c C = + + c All of these moves re vlid for rnpds. We lso hve two new locl moves for rnpds tht do not exist for cprns: 5

6 Antenn Jumping We define n ntenn to e n interior oundry vertex of degree one. If we hve such vertex nd it is djscent to vertex v, then ntenn jumping is performed y jumping the ntenn over n edge incident to v nd into different fce. The conductnces re unchnged y this move. c c Antenn Asorption e c g f E B C F A d A = g + + c + g g B = + + c + g cg C = + + c + g D D = d c + g c E = e c + g c F = f c + g Conjecture 1. Any two electriclly equivlent grphs ssocited to rnpds re connected y the locl moves stted in this section. 4 Medil Grphs nd Z-sequences Defnition 12 ([1]). For n cprn Γ with n oundry vertices, v 1,..., v n clockwise, plce 2n points, t 1,..., t 2n, round the oundry of the disk in clockwise order such tht for even i, t i 1 nd t i re on either side of v i/2. Tht is, etween v j nd v j+1 we hve, in clockwise order, v j, t 2j, t 2j+1, v j+1 where indices re modulo n for the v i s nd modulo 2n for the t i s. For ech edge e in Γ, let m e e its midpoint. Then the medil grph, M(Γ), of Γ is the grph with vertices {t i 1 i 2n} {m e e E} nd edges {(m e, m f ) e, f edges in Γ on the sme fce nd shring vertex} {(m e, m e ) m e spike into fce of Γ} {(t i, m e ) t i, e on the sme fce, i odd, e n edge connected to v (i+1)/2 } {(t i, m e ) t i, e on the sme fce, i even, e n edge connected to v i/2 }. Exmple 2. The medil grph of the cprn in Figure 1 is drwn elow. 6

7 Remrk 2. For n edge e of cprn Γ, m e hs degree 4 in M(Γ). This mens tht if we pproch m e vi some edge in M(Γ), we cn turn right, turn left, or go stright. Defnition 13. We define n equivlence reltion on the edges of M(Γ) s follows. For every e n edge of Γ with corresponding vertex m e in M(Γ), let the four medil edges djcent to m e e v e,1, v e,2, v e,3, nd v e,4 in clockwise order. Then v e,1 v e,3 nd v e,2 v e,4. Then the medil strnds of M(Γ) re the equivlence clsses of edges of M(Γ) induced y. In other words, medil strnd is pth in M(Γ) where we lwys go stright through vertex m e. We frequently smooth the corners of strnd in depictions of the medil grph. Exmple 3. The medil grph depicted in Exmple 2 hs two medil strnds, s shown elow. Remrk 3. Ech vertex t i hs degree 1 in M(Γ). This mens tht ech t i is n endpoint of exctly one medil strnd. Defnition 14. [1] Let Γ e cprn with n oundry vertices, v 1,..., v n clockwise, nd let t 1,..., t 2n e the endpoints of medil strnds. Lel the strnd eginning t t 1 s with 1. The remining strnds re leled 2 through n so tht if i < j, the endpoints of the strnd i re t nd t, nd the endpoints of strnd j re t c nd t d, we hve either < c, d or < c, d. In other words, the first endpoint of strnd i ppers clockwise from t 1 efore the first endpoint of j. For ech i {1, 2,, 2n} let z i e the numer ssocited with the strnd with n endpoint t t i. The sequence z 1,..., z 2n is clled the z-sequence of Γ. We cn now mke some nlogous definitions for rnpds. Defnition 15. For n rnpd Γ we cn define the medil grph M(Γ) of Γ to e s for cprn, treting the interior oundry vertex s n internl vertex. We otin medil strnds s in the cprn cse. Exmple 4. The medil grph of the rnpd in Figure 1 is 7

8 Defnition 16. For n rnpd Γ, we cn define the z-sequence y eginning with the construction for cprns to get some permuttion of the multiset {1, 1, 2, 2,..., n, n}. For ech i, lel one strnd endpoint of s i s s + i nd the other s i such tht the strnd from s i to s + i moves clockwise round. Additionlly, if strnd contins self-intersection, put r over its endpoints lels. The z-sequence of Γ is the sequence of lels of strnd endpoints, strting t t 1 going clockwise. Exmple 5. Consider the rnpd from Exmple 4. If the oundry vertex on the left is v 1 nd the oundry vertex on the right is v 2, then the z-sequence for this rnpd is 1 +, 2, 2 +, 1. Defnition 17. A motion on tringulr fce ABC of medil grph is defined y the following (locl) trnsformtion: Remrk 4. Medil grph motions correspond directly to Y moves, so the grphs of two rnpds re Y equivlent if nd only if their medil grphs re equivlent y motions. Defnition 18. Suppose s nd t re segments of two medil strnds s nd t such tht s nd t oth hve endpoints nd, s nd t do not intersect themselves or ech other etween nd, nd the medil edges in s nd t djcent to nd ut not on s nd t re outside the region enclosed y s nd t. Define medil lens to e the region enclosed y s nd t etween nd. Defnition 19. Define medil loop to e the region enclosed y medil strnd segment whose endpoints re the sme. Defnition 20. Define medil circle to e medil strnd tht does not hve endpoints. Exmple 6. In the following exmple medil grph, the green strnd is medil circle. It ounds two lenses with the lue strnd. The red strnd ounds medil loop. The lue nd red strnds ound medil lens. 8

9 With these definitions in hnd, we my stte the first of two min theorems of this section. Theorem 4. An rnpd is irreducile y Y moves, series reductions, prllel reductions, pendnt removl, self-edge removl, nd ntenn jumping if nd only if the following conditions ll hold: () M(Γ) contins no medil circles. () M(Γ) contins t most one self-intersecting strnd. Furthermore, if such strnd exists, it intersects itself only once, with the loop it cretes contining the interior oundry vertex. (c) Any two distinct medil strnds of M(Γ) intersect t most twice. Furthermore, if two distinct medil strnds intersect twice, creting lens, the lens they crete contins the interior oundry vertex. We first prove the ckwrds direction. Proof. Edges in series nd prllel induce medil lens fce in the medil grph. Pendnts nd selfedges induce medil loop fce in the medil grph. Thus, no grph stisfying () nd (c) cn e reduced only y series reductions, prllel reductions, self-edge removl, nd pendnt removl. Medil circles remin medil circles under medil grph motions nd ntenn jumping. Similrly, medil lenses nd medil loops not contining remin medil lenses nd medil loops not contining under motions nd ntenn jumping. Thus, if grph stisfies properties (), (), nd (c), it will lwys stisfy the properties fter Y moves nd ntenn jumping, nd hence will never e le to e reduced y series reductions, prllel reductions, pendnt removl, or self-edge removl. To prove the forwrds direction, we require severl lemms. Lemm 1. In the medil grph of n rnpd irreducile y Y moves, series reductions, prllel reductions, pendnt removl, nd self-edge removl: () Any medil lens contins the interior oundry vertex. () Any medil loop contins the interior oundry vertex. (c) Any medil circle contins the interior oundry vertex. Proof. Suppose our medil grph hs medil lens. We follow the lgorithm in Lemm 6.2 of [1], noting tht we my still perform motions on ny tringulr fce in the medil grph tht does not contin the oundry vertex. For lens tht does not contin the oundry vertex, none of its fces contin the oundry vertex. Following the lgorithm in Lemm 6.2 will never cuse the lens to contin new vertices. Thus, for rnpds, ny medil grph with medil lens not contining the oundry vertex is equivlent y motions to one with lens (still not contining the oundry vertex) s fce. We cn pply the sme rgument for medil loops, in which cse we otin medil loop fce, nd medil circles, in which cse we otin medil circle fce. Lens fces my e reduced y series or prllel trnsformtion (depending on whether vertex is inside the lens) nd medil loop fces my e reduced y pendnt removl or loop removl (depending on whether vertex is inside the loop). In oth cses, we hve contrdiction ecuse our grph ws reduced. Medil circle fces cnnot exist. 9

10 Lemm 2. Let p, q, r e pirwise-intersecting medil strnds in n rnpd Γ irreducile y Y moves, series reductions, prllel reductions, pendnt removl, nd self-edge removl. Define v qr e n intersection of strnds q nd r nd define v pr, nd v pq similrly. Suppose tht the segments of strnds p, q, nd r etween v pr nd v pq, v pq nd v qr, nd v pr nd v qr form the sides of tringle T. Tht is, these strnd segments do not intersect themselves or ech other except for t v qr, v pr, nd v pq. If T does not contin, then we cn pply sequence of motions on fces of T to otin network where T is fce of the medil grph. Proof. We prove this inductively on the numer of fces inside T. If T only hs one fce, no motions re required. If T hs more thn one fce, some strnd s intersects the oundry of T twice. If these intersections occur on the sme strnd, there would e lens inside T. Since T does not contin, this contrdicts Lemm 1. So, without loss of generlity, we cn ssume these intersections occur on strnds q nd r nd tht the intersection points re v qs nd v rs. Then q, r, nd s enclose region U with vertices v qr, v qs, nd v rs. Since U is strictly inside T, U hs fewer fces thn T. Thus, y induction, we cn perform motions so tht there re no intersections inside U. These motions will not increse the numer of fces in T. Next we cn perform motion to move U outside T, decresing the numer of fces in T y one. By induction, we cn perform nother sequence of motions to mke T fce of the medil grph. Lemm 3. In n rnpd irreducile y Y moves, series reductions, prllel reductions, pendnt removl, self-edge removl, nd ntenn jumping with no medil circles, there exists t most one medil loop. Tht is, there exists t most once medil strnd with self-intersection, nd if such strnd exists, it intersects itself only once. Proof. For the ske of contrdiction, suppose we hve t lest two medil loops. Let l e medil loop contins no other medil loops within its interior. By the previous lemm, the interior oundry vertex must e inside l. Clim 1: We cn use motions to mke l fce in the medil grph. Any strnd segment s in l with oundry on l contins no self-intersection. Then it divides l into lens nd tringle T. The lens must then contin the interior oundry vertex, so T does not. By Lemm 2 we cn use motions to mke T empty. These motions do not increse the numer of strnd segments in l. After we mke T empty, motion on T decreses the numer of strnd segments in l. Applying this process repetedly, we cn mke l fce, proving Clim 1. By Clim 1, up to electricl equivlence, we my ssume l is fce in the medil grph. This fce must correspond to the vertex of the rnpd y prt () of Lemm 1. Since every medil loop contins, every medil loop contins l. Without loss of generlity, let l e such tht l is the only loop contined in l. Clim 2: We cn use motions to mke l contin only l. Let the self-intersection of l e. As in the proof of Clim 1, ny strnd segment s in l with endpoints s 1 nd s 2 on l contins no self-intersection. Since s cnnot intersect l or crete lens, s 1, s 2, nd crete tringle T tht does not contin. By Lemm 2 we cn use motions to mke T empty. These motions do not increse the numer of strnd segments in l. After we mke T empty, motion on T decreses the numer of strnd segments in l. Applying this process repetedly, we cn mke l contin only l, proving Clim 2. Now we re in one of three situtions, shown elow. 10

11 Cse 1A: Cse 1B: Cse 2: In cses 1A nd 1B, ntenn jumping yields medil lens not contining (note the prllel edges in 1A nd the prllel edges fter Y move in 1B). In cse 2, ntenn jumping yields medil loop not contining (note the self-edge). This is contrdiction with Lemm 1, so we hve proven the lemm. To prove the next Lemm, we need to introduce some nottion. Defnition 21. Let Γ e grph with vertex set V. Let V V, V e ll vertices in V or djcent to V, E e the edges of Γ with oth endpoints in V, nd E e the edges of Γ with t lest one endpoint in V. Then the strong restriction of Γ to V is Γ = (V, E ) nd the wek restriction of Γ to V is Γ = (V, E ). The oundry vertices of Γ re (B V ) {v V v is djcent to vertex in V \V }. The oundry vertices of Γ re (B V ) (V \ V ). Remrk 5. Both the strong restriction nd wek restrictions of Γ to V re sugrphs of Γ. Remrk 6. All grphs Γ re equl to the strong restriction to their vertex set. Assuming connectedness, they re lso equl to the wek restriction to their interior vertex set. Lemm 4. The medil grph of n irreducile rnpd does not contin ny medil circles. Proof. For the ske of contrdiction, suppose there exists n irreducile rnpd Γ whose medil grph M(Γ) contins medil circle c. Without loss of generlity, ssume c contins no medil circles in its interior. Let Γ e the strong restriction of Γ to the vertices inside c. Then M(Γ ) is the restriction of M(Γ) to medil edges strictly contined in c. Note tht M(Γ ) contins no medil circles nd t most one medil loop. We consider severl disjoint cses: Cse 1: The medil circle c contins no self-intersection. If strnd segment inside c does not hve self-intersection, then the strnd cretes two lenses with c. Only one of those lenses cn contin the oundry vertex, so y Lemm 2 we find Γ is not reduced. This is contrdiction, so every strnd segment inside c with endpoints on c contins self-intersection. Since M(Γ ) contins t most one medil loop, c contins t most one strnd segment. The grph is connected, so c cn t e empty. The medil grph contining only c nd medil loop s segments is shown elow. 11

12 This is not possile configurtion s its network is reducile. Cse 2: The medil circle c contins exctly one self-intersection. The medil circle c cnnot form figure-eight, s tht cretes two disjoint loops. Thus it must hve the following form: Just s in cse 1, ny strnd segment inside c without self-intersections cretes lens not contining. From erlier, c cn strictly contin t most one medil loop which cn e its only strnd. This yields three configurtions up to motions for strnds in c: The first two re reducile. The third cn t exist, s the medil grph of connected grph is connected nd c is not the full medil grph of ny rnpd. Cse 3: The medil circle c contins t lest two self-intersections. 12

13 Let p e point on c. Then without loss of generlity, moving long the to rech the first intersection e 1 of the strnd segment, we crete counterclockwise loop. This must contin. Then, continuing long c to the second intersection e 2 we cnnot go clockwise, s tht cretes lens not contining. Thus we must go counterclockwise, nd segment of c mus look s elow. e 1 e 2 p Now, let l e the shown strnd segment of c. Then ny not-self-intersecting strnd segment s in l intersecting the oundry of l t endpoints s 1 nd s 2 either cretes lens with l not contining or tringle T with s 1, s 2, nd e 2. The first cse is reducile. In the second we my pply motions on regions in T to mke T empty. Then motion inside T decreses the numer of strnd segments intersecting l. Thus, we my ssume every strnd segment s in l contins self-intersection. Since the loop with self-intersecting vertex e 1 is strictly contined in c, no other loops strictly in c exist. Thus, no strnds my intersect l. But this is then reducile. Lemm 5. In n rnpd irreducile y Y moves, pendnt removl, self-edge removl, prllel reductions, series reductions, nd ntenn jumping, ny two medil strnds intersect t most twice. Proof. Assume for contrdiction we hve two medil strnds p nd q tht intersect (t lest) three times. Without loss of generlity, ssume p contins no self-intersection. As we move long q let our three intersections e v 1, v 2, nd v 3. Then p nd q form two lenses - one with endpoints v 1 nd v 2, nd one with endpoints v 2 nd v 3. Since these lie on opposite sides of p, they hve disjoint interiors. Thus t lest one medil lens does not contin the interior oundry vertex, contrdiction. With this series of lemms, we cn now prove Theorem 4. Proof. Let Γ e n rnpd irreducile y our moves. Prt () holds y Lemm 4. Prt () nd Lemm 3 imply tht M(Γ) contins t most one medil loop. By Lemm 1, if there is medil loop, then it must contin the interior oundry vertex. This proves prt () of the theorem. Finlly, y Lemm 5, ny two distinct strnds of m(γ) intersect t most twice. Lemm 1 tells us tht ny two strnds tht do intersect twice must contin the interior oundry vertex in the lens they crete, so we hve proven prt (c) or the theorem. Conjecture 2. A grph is irreducile y Y moves, pendnt removl, self-edge removl, prllel reductions, series reductions, ntenn jumping, nd ntenn sorption if nd only if it hs three medil strnds which pirwise intersect twice, there is medil strnd tht self-intersects, nd it is irreducile y Y moves, pendnt removl, self-edge removl, prllel reductions, series reductions, nd ntenn jumping. Theorem 5. Two rnpds irreducile y Y moves, pendnt removl, self-edge removl, prllel reductions, series reductions, nd ntenn jumping hve the sme z-sequence if nd only if their grphs relted y Y moves. 13

14 Proof. Note tht motions don t chnge the reltive order of strnd endpoints, or whether strnds go round the interior oundry vertex clockwise or counterclockwise, so the z-sequence is preserved y motions nd thus Y moves. For the other direction, we induct on the numer of strnds. If only one strnd exists, the theorem holds trivilly. Suppose we hve we hve two rnpds Γ nd Γ with multiple strnds nd the sme z-sequence. Ech strnd in Γ which does not intersect itself divides Γ into two regions. For strnd i without selfintersection, cll the region which does not contin the oundry vertex Γ i,1, nd the other Γ i,2. Choose s such tht Γ s,1 contins the fewest possile numer of regions. We cn define Γ s,1, nd Γ s,2 s we did for Γ. Since we chose s where Γ s,1 is miniml, every strnd in region Γ s,1 intersects s. Let G e the grph we otin when we perform Y moves on Γ to minimize the numer of regions on the side of strnd s not contining the interior oundry vertex. Define G 1 nd G 2 s the regions prtitioned y strnd s in G where G 1 is the side corresponding to Γ s,1. Define G, G 1, nd G 2 similrly from Γ. We clim tht G 1 contins no medil strnd intersections. Suppose for contrdiction tht p nd q re strnd segments tht lie within G 1 nd intersect t v 0 in G 1. Let the endpoint of p on s e v 1. Let the endpoint of q on s e v 2. Let the tringle formed y v 0, v 1, nd v 2 e T. By motions on tringulr fces in T, we my mke T empty without incresing the numer of regions in G 1. Then, motion on T moves T to G 2, decresing the numer of regions in G 1 y one, contrdiction. Similrly, G 1 contins no medil strnd intersections. Furthermore, G strongly restricted to vertices in G 2 is n irreducile grph whose z-sequence my e computed from from the z-sequence of G y deleting strnd s nd releling the remining strnds. Since we my determine from the z-sequence of G which strnd segments of G 1 intersect s, G wekly restricted to vertices G 1 nd G wekly restricted to vertices in G 1 re isomorphic grphs. As ove, G strongly restricted to G 2 is grph irreducile y Y moves, pendnt removl, self-edge removl, prllel reductions, series reductions, nd ntenn jumping whose z-sequence my e computed from from the z-sequence of Γ y deleting strnd s nd releling the remining strnds. Then G 2 nd G 2 hve the sme z-sequence. By the inductive hypothesis, G 2 nd G 2 re Y equivlent. This mens G nd G, nd therefore Γ nd Γ, re Y equivlent s well. 5 Conditionl Locl Moves nd Spider Grphs 5.1 Spider Grphs Defnition 22. The stndrd grphs, denoted y Σ n for n 2, re certin criticl cprns with z-sequence 1, 2,..., n, 1, 2,..., n. See Section 7 of [1] for detiled construction. In [2], it ws shown tht ech Σ 4m+3 is recoverle. An lgorithm tht trnsforms ny criticl cprn into some Σ 4m+3 ws introduced in [1] nd ws centrl to the proof tht criticl cprns re recoverle. We define fmily of cprns tht re electriclly equivlent to stndrd grphs. These grphs will e clled 4-periodic grphs. To do this, we ll first need to define some specil types of edges. Defnition 23. An edge is oundry edge if it connects two oundry vertices. An edge is oundry spike if it is connected to oundry vertex of degree 1. We will denote the 4-periodic grphs s Π n for n 3. The construction is s follows: For n 3, we strt with n n-gon whcich we will consider to e lyer 1. Then we extend our polygon to hve exctly n+1 4 lyers, where ech lyer is concentric n-gon connected to the n-gon from the previous lyer y n edges etween respective vertices (see Fig. 1). For n 3 mod 4 we will hve n+1 4 lyers. For n 0 mod 4 we will hve n 4 lyers nd dd n 2 consecutive oundry spikes. For n 1 mod 4, we will hve n 1 4 lyers nd dd ll n oundry spikes. For n 2 mod 4, we will hve n 2 4 lyers, dd ll n oundry spikes, nd then connect n 2 consecutive oundry vertices. In ech cse, we let ll outermost vertices e oundry vertices (see Figure 2). 14

15 Π 3 Π 4 Π 5 Π 6 Π 7 Π 8 Figure 2: The first six 4-periodic cprns, Π n for 3 n 8. Defnition 24. In Π n for n even, we cn see tht s we go round the oundry of the disk, we hve set of oundry vertices with oundry spikes nd then set of oundry vertices connected y oundry edges. Let the first nd lst of the oundry vertices connected y oundry edges e v nd w. For n 0 mod 4, ech of these vertices is of degree 2. We will refer to the edges connected to v nd w tht re not oundry edges s oundry pseudo-spikes. For n 2 mod 4, ech of these vertices is of degree 3. The vertices hve oundry edge, n edge going inwrd, nd n edge connected to n internl vertex tht is connected to oundry spike. We will refer to the lst of these edges s pseudo-oundry edge. Lemm 6. Let Π n e 4-periodic grph nd let l e the numer of lyers in this grph. Let s e medil strnd in Π n with endpoints s 1 nd s 2 such tht the center fce of the grph is to the right of the strnd when going from s 1 to s 2. Then the numer of strnd endpoints etween s 1 nd s 2 to the left of the strnd is: 1. 4l 2 if n 3 mod l 1 if n 0 mod l if n 1 mod l + 1 if n 2 mod 4. Proof. Suppose the lemm is true for cse (3) for grph with l 1 lyers. This mens tht given ny strnd s, the numer of strnd endpoints etween s 1 nd s 2 is 4l 4 in such grph. Choose 15

16 Figure 3: Types of edges in Π 4, Π 6, nd Π 8. Boundry edges pper in purple, oundry spikes in pink, pseudo-oundry edges in light lue, nd oundry pseudo-spikes in ornge. ny strnd s. This strnd prtitions the grph into two pieces. One of these pieces contins most of the center fce. Add extr vertices nd edges to the grph so tht the grph still hs l 1 lyers, ut the underlying polygon now hs 4 more sides. We will ssume without loss of generlity tht these vertices nd edges re dded in the piece of the grph tht contins most of the center fce. This mens it won t ffect s t ll. Now we dd lyer to get to the next grph tht flls under cse (3). We cn oserve tht dding the lyer forces us to hve two extr sides of the polygon etween s 1 nd s 2, which corresponds to exctly 4 extr strnd endpoints (see Figure 4). Thus, the numer of strnd endpoints etween s 1 nd s 2 is 4l = 4l, s desired, nd we hve proven cse (3) of the lemm. The proof for cse (1) is similr. Figure 4: Adding lyer dds two extr oundry spikes etween strnd endpoints. Now suppose tht n 0 mod 4. We clim tht ech strnd must hve one end crossing oundry spike nd the other end crossing n oundry edge or pseudo-oundry edge. To prove this, suppose for contrdiction tht the strnd s strts y crosses oundry spike on oth ends (the other cse is similr). Then the strnd looks exctly like strnd from cse (3) nd there must e 4l strnd endpoints etween s 1 nd s 2 to the left. This mens tht there re t lest 4l oundry spikes in the grph (4l 2 internl strnds nd 2 spikes tht S crosses). Since n 0 mod 4, we get the following reltion: 4l 2 2 n = 2l + 1 = = 2 n = n + 2 > n 2 2 This is contrdiction, since the numer of oundry spikes in such grphs is exctly n 2. Now we will prove tht in ny such grph, ny strnd tht hs one end crossing oundry spike nd one 16

17 end crossing oundry edge or pseudo-oundry edge hs exctly 2l 1 strnd endpoints etween its endpoints. Assume tht the lemm holds for grph with l 1 lyers. Let s e strnd in the grph. We cn increse the size of the polygons in ech lyer s we did for the proof of cse (3) without ffecting s. Agin, dding lyer dds exctly two sides of the polygon etween s 1 nd s 2, nd consequently 4 more strnd endpoints. The proof for cse (4) is similr, where we use oundry edges insted of oundry spikes nd oundry spikes nd oundry pseudo-spikes insted of oundry edges nd pseudo-oundry edges. Corollry 1. For n 3, Π n is electriclly equivlent to Σ n. Proof. Let n 1 mod 4. We first note tht the numer of medil strnds in Π n is n. Also, the numer of endpoints is 2n, s every oundry vertex corresponds to exctly two endpoints. Tke ny strnd s with endpoints s 1, s 2. By Lemm 6, we know tht the numer of strnds etween s 1 nd s 2 is 4l on one side. Since n 1 mod 4, we know tht 4l = 4 n+1 4 = 4 n 1 4 = n 1, so there re n 1 strnd endpoints etween s 1 nd s 2. Since there re 2n 2 endpoints other thn s 1 nd s 2, it follows tht the numer of endpoints etween s 1 nd s 2 on the other side is lso n 1. Since s ws n ritrry strnd, the grph hs the sme z-sequence s the stndrd grph. Also note tht the medil grph is lenseless. This mens Π n is electriclly equivlent to Σ n. The cses when n 2 mod 4, n 3 mod 4, nd n 0 mod 4 re similr. Defnition 25. A spider grph, Ξ n, is defined for ech n 3. Ξ n is constructed y plcing oundry vertex inside the center fce of Π n, nd connecting it to ech vertex on the oundry of tht fce. Ξ 3 Ξ 4 Ξ 5 Ξ 6 Ξ 7 Ξ 8 Figure 5: The first six spider grphs, Ξ n for 3 n 8. 17

18 5.2 Conditionl Locl Moves In ddition to the 7 locl moves descried in section 3, there re lso conditionl locl moves we cn perform on rnpds. These re similr to the locl moves in tht they do not chnge the response mtrix, ut the formuls re not sutrction free. Thus, we cn only perform the conditionl locl moves s long s they do not give us ny negtive conductnces. We hve the following conditionl locl moves (see Appendix A for formuls): Tringle Conditionl Locl Move c e f d B C A E F D Squre Conditionl Locl Move c e f d g h i B C A E F D G H I Pentgon Conditionl Locl Move c d e f g h i j k l m A B C D E F G H I J K L M Conjecture 3. We conjecture hexgon conditionl locl move s well: 18

19 Q o j p O J P i h c d e f k l I H D C B E F A K L n g m N G M q We hve conjecturl formuls for this move (see Appendix A). We were le to check tht these formuls preserve every entry of the response mtrix except for the entry in the row nd column corresponding to the interior oundry vertex. The computtions were too lrge to check for this entry. However, we did check tht the response mtrices were the sme for severl choices of vlues for the vriles in one of the networks. Conjecture 4. Notice tht ech of the conditionl locl moves listed ove is move on sugrph of Π n. We conjecture tht there exists conditionl locl move for ech n 3 where pplying the locl move on Π n does the following: For n 3 mod 4, we chnge oundry edge from the outermost lyer into oundry spike on the opposite side of the grph. For n 0 mod 4, we chnge oundry spike from the outermost prtil lyer into oundry spike on the opposite side of the grph. For n 1 mod 4, we chnge oundry spike from the outermost prtil lyer into oundry edge on the opposite side of the grph. For n 2 mod 4, we chnge oundry edge from the outermost prtil lyer into oundry edge on the opposite side of the grph. It lso seems tht the formuls for these moves hve comintoril interprettion in terms of groves or grove-like structure (see [7]). For exmple the numertor in the formul for G, H, nd I in the pentgon conditionl locl move is sum of groves. 5.3 Properties of Spider Grphs Defnition 26. Let P, Q e ordered susets of B in the network Γ. As in [1], we define Λ(P ; Q) to e the sumtrix of Λ(Γ) with rows corresponding to elements of P nd columns corresponding to n elements of Q. Lemm 7. Assume Γ e spider grph with oundry vertices B nd interior oundry vertex. Let P, Q = {p 1,, p k }, {q 1,, q k } e susets of the B {} such tht p 1, p 2,, p k, q k, q 2, q 1 re in clockwise order. If (P, Q) is connection, s defined in section 2, then ( 1) k det Λ(P ; Q) > 0. Otherwise, det Λ(P ; Q) is zero. Proof. This lemm hs een proven for cprns in Theorem 4.2 of [1]. Since the disjoint pths in the connection (P, Q) cnnot pss through oundry vertices nd we hve restricted P nd Q to not 19

20 contin, we re not le to use the center fce of the spider grph in our pths etween P nd Q. We therefore hve the sme set of possile connections s the 4-periodic grphs do. Other thn properties true for ll plnr grphs, the proof of Theorem 4.2 relied only on the signs of the permuttions for ll possile pths etween P nd Q. Since these permuttions re the sme for spider grphs nd 4-periodic grphs cprns the sme proof holds. Theorem 6. Let Ξ n e spider grph. Consider the z-sequence for Ξ n. Let S = (s +, s ) e n ritrry strnd. Then if n is odd, then there re exctly n 3 strnd endpoints etween s + nd s, where we lwys mesure the distnce moving clockwise. If n is even, then for ll ut two strnds, there re exctly n 3 strnds etween s + nd s. One will hve n 2 strnds etween its endpoints, nd the other will hve n 4. Proof. First ssume n is odd. We note tht for odd n, ech strnd in the medil grph of Ξ n will first cross oundry spike or oundry edge nd end y crossing the sme type of edge. We know tht the numer of strnd endpoints etween s + nd s in Π n is exctly n 1. Note tht dding oundry vertex nd edges in the middle of the grph forces us to hve two more strnd endpoints etween s nd s + (see Figure 6). Therefore, there re two fewer strnd endpoints etween s + nd s nd the numer of strnds etween s + nd s is n 3. When n 0 mod 4, we hve n 2 1 oundry edges, n 2 oundry spikes, nd 2 pseudo-oundry edges. When n 2 mod 4 we hve n 2 oundry edges, n 2 1 oundry spikes, nd 2 oundry pseudo-spikes. Recll tht in every Π n for even n, ech medil strnd hs one of its endpoints crossing oundry edge or pseudo-oundry edge nd nother crossing oundry spike or oundry pseudo-spike. When we dd the str grph in the center to get Ξ n, we get two dditionl strnd endpoints etween ech s nd s +, s in the odd cse. This mens tht from the viewpoint of s, s + is shifted y exctly one oundry vertex. Thus, ll ut two strnds will still hve one endpoint crossing oundry edge or pseudo-oundry edge nd the other crossing oundry spike or oundry pseudo-spike. As in the odd cse, there re now two more strnd endpoints etween s + nd s, nd re therefore n 3 strnd endpoints etween s + nd s. We now look t the two strnds tht do not hve this property. First, let n 0 mod 4. One of these strnds hs oth its endpoints crossing the pseudo-oundry edges (cll this s 1 ), so there re n 2 strnd endpoints etween s + 1 nd s 1. The other strnd hs oth its endpoints crossing the rightmost nd leftmost oundry spikes (cll this s 2 ), so there re n 4 strnd endpoints etween s + 2 nd s 2. Now, let n 2 mod 4. One of these strnds hs oth its endpoints crossing the oundry pseudo-spikes (cll this s 1 ), so there re n 2 strnd endpoints etween s + 1 nd s 1. The other strnd hs oth its endpoints crossing the rightmost nd leftmost oundry edges (cll this s 2 ), so there re n 4 strnd endpoints etween s + 2 nd s 2. Figure 6: Adding the str grph to the center dds two more strnd endpoints etween s nd s +. 20

21 6 Recoverility Conditions 6.1 Recoverle Rnpds from Criticl Cprns Lemm 8 (Lemm 11.3 from [1]). Let Γ e criticl cprn. Then Γ hs oundry edge or oundry spike. Lemm 9 (Lemm 11.1 from [1]). Let Γ e criticl cprn. Then Γ remins criticl fter deleting oundry edge. Lemm 10 (Lemm 11.2 from [1]). Let Γ e criticl cprn. Then Γ remins criticl fter contrcting oundry spike. Lemm 11. Let Γ e n rnpd with interior oundry vertex, nd let e e oundry edge or oundry spike. If deleting or contrcting e = pq reks some connection (P, Q) s.t. / P Q, then we cn derive the conductnce of e from Λ(Γ). Proof. Corollries 4.3 nd 4.4 of [1] prove this for the cprn cse. However, this remins true for rndps if we restrict P nd Q to not include the interior oundry vertex. Defnition 27. A str grph is centrl oundry vertex with only oundry spikes or pendnts ttched. Inserting str grph into fce of grph mens dding oundry vertex inside the fce nd connecting it to ny numer of vertices on the fce. Exmple 7. Consider the following grph: One wy to insert str grph into the upper fce is illustrted elow: Theorem 7. Assume Γ is criticl cprn. If Γ e the rnpd otined from inserting str grph into ny fce of Γ, then Γ is recoverle. Proof. Let Γ = (V, E) nd S = (V S, E S ) e the str grph tht ws inserted into Γ to otin Γ. Assume we re given Λ(Γ ). First, note tht (V \ {}, E \ E S ) = Γ, which is known to e criticl. So, y Lemm 8, we lwys hve oundry edge or oundry spike with which to egin the following process. First, pick oundry edge or oundry spike, e, to delete or contrct. Notice tht deleting or contrcting e must rek connection (P, Q) in Γ y the definition of criticl. Since pths in connections cnnot pss through other oundry vertices, dding ck in S does not repir the roken 21

22 connection, nd (P, Q) is roken in Γ. Thus, y Lemm 11, we know the conductnce of e. Knowing this llows us to derive the response mtrix of the grph fter contrcting or deleting e, which we will cll Γ (see section 8 of [1]). Now, y Lemms 9 nd 10, our Γ without S is still criticl. Then, we cn continue this process, deleting nd contrcting oundry edges nd spikes one y one, until we re left with S. In other words, we will eventully know the conductnce of every edge in Γ except for those in G, ut will know the response mtrix of G. However, t tht point, S will e cprn with only oundry vertices, nd hence its response mtrix is exctly its Kirchhoff mtrix. Therefore, Γ is recoverle. Exmple 8. Note the Ξ n is Π n with str grph inserted into fce. So, ll the spider grphs re recoverle. Exmple 9. By Theorem 5, ll irreducile grphs tht hve the sme z-sequence s spider grph re recoverle. 6.2 Necessry Condition for Recoverlilty of Rnpds Before stting the necessry condition, we first need to define the following lgorithm for otining cprn from certin type of rnpd. Suppose we re given n rnpd with the interior oundry vertex. Algorithm Turn ll the neighors of into oundry vertices. 2. Remove from the grph (with ll the edges incident to it). 3. If the resulting grph is cprn, stop. 4. Otherwise, pply the ove steps to ech oundry vertex (in ny order) tht cn t e plced on the disk until it results in cprn. Recll tht in ny rnpd, the interior oundry vertex hs to e inside polygon (one of the fces of cprn). Theorem 8. Assume Γ is n rnpd where the internl oundry vertex is the center of str grph in the interior of some fce f of the cprn Γ. Apply Algorithm 1 itertively strting t the internl oundry vertex, nd cll the resulting cprn Γ. Then Γ is criticl if Γ is recoverle. Proof. Assume Γ is not criticl. Then the numer of edges cn e reduced. Note tht Γ is isomorphic to sugrph of Γ, ut not necessrily to sunetwork (not respecting the type of vertices). So Γ my hve more oundry vertices. Hving more oundry vertices my only decrese the numer of possile moves. This implies tht the numer of edges in Γ could e reduced to egin with, nd thus Γ is not recoverle. Acknowledgements This reserch ws crried out s prt of the 2018 REU progrm t the School of Mthemtics t University of Minnesot, Twin Cities. The uthors re grteful for the support of NSF RTG grnt DMS They would lso like to thnk Sunit Chepuri nd Pvlo Pylyvskyy for their guidnce nd encourgement, Eric Stucky for his mny helpful comments on the mnuscript, nd Alexndr Emry, Sylvi Frnk, nd John O Brien for their insights. 22

23 Appendix A Formuls for Conditionl Locl Moves Here we list the conditionl locl moves with formuls. Tringle Conditionl Locl Move e c f E B C A d F D = = c = d = e = f = AB + AD + AE + AF + BD B + D + E + F BF B + D + E + F BC + BE + CD + CE + CF B + D + E + F DF B + D + E + F EF B + D + E + F DE B + D + E + F Squre Conditionl Locl Move A = B = C = D = E = F = e f e (f + de + df + ef) de cd f d f + de + df + fe e f + de + df + fe d f + de + df + fe f g G I f c d h F C B D A H i e E In this cse the move is symmetric, so we only include formuls for one direction. A = B = g + eg + fg + gi + eg def g( + e + f + i) i + e + f + i 23

24 ch + fh + ceh + cfh + chi def C = h( + e + f + i) D = E = F = G = H = I = d(gh + def + efg + efh + egh + fgh + ghi) gh( + e + f + i) ei + e + f + i fi + e + f + i gh + def + efg + efh + egh + fgh + ghi h( + e + f + i) gh + def + efg + efh + egh + fgh + ghi g( + e + f + i) gh + def + efg + efh + egh + fgh + ghi ef Pentgon Conditionl Locl Move m g h c d i e j G H C D B I E A J l f k L F K M Let α = ABM +AF M +AGM +ALM +BF M +BJM +BKM +F GM +F JM +F KL+F KM + F LM +GJM +GKM +JLM +KLM, β = F KL+F KM +F LM +KLM, γ = fgj(d+h+i+m), nd δ = hi + fhi + ghi + fhi + hij + fghi + fhij + ghij. = = c = C + d = D + e = E + A(β + BKM + GKM) α B(β + ALM + JLM) α ABGIM + AF GIM + BF GIM + BGIJM + BGIKM DF GJM Iα D(DF GJM + HF GJM + IF GJM) IHα ABHJM + AF HJM + AGHJM + AHJLM + BF HJM DF GJM Hα 24

25 (β + BKM + GKM)(β + ALM + JLM) f = LKα g = h = H + i = I + j = k = l = m = A = + B = + C = c + G(β + ALM + JLM) α DF GJM + HF GJM + IF GJM Iα DF GJM + HF GJM + IF GJM Hα J(β + BKM + GKM) α β + BKM + GKM F L β + ALM + JLM KF HIα + DF GJM + HF GJM + IF GJM F GJM D = dfgjm γ E = e + F = G = g + (δ + hil + hijl) γ (δ + hik + ghik) γ g(dfhj hi fhi fhi hij ihk) γ j(dfgi hi fhi ghi hil fhi) γ f(γ + δ + hil + hijl)(γ + δ + hik + ghik) γ(γ + δ + hil + hik + fhik + fhil + ghik + hijl + hikl) H = fghjm γ I = fgijm γ J = j + K = L = M = g(δ + hik + ghik) γ j(δ + hil + hijl) γ k(γ + δ + hil + hijl) γ + δ + hil + hik + fhik + fhil + ghik + hijl + hikl l(δ + hik + ghik) γ + δ + hil + hik + fhik + fhil + ghik + hijl + hikl ghikl γ + δ + hil + hik + fhik + fhil + ghik + hijl + hikl 25

26 Conjectured Hexgon Conditionl Locl Move Q o j p O J P i h c d e f k l I H D C B E F A K L n g m N G M q In this cse the move is symmetric, so we only include formuls for one direction. Let α = gmn + gmq + gnq + mnq, β = q + gq + hq + nq + gq + lq + mq + ghq + glq + hlq + hmq + lnq, γ = deghlq + dghjlq + dghklq + eghilq + eghjlq + ghijlq + ghiklq + ghijklq, nd δ = dghlpq + ghilpq + eghloq + ghkloq + ghjloq + ghjlpq + ghlopq. A = B = C = c + D = d + E = e + F = f + G = H = I = i + J = K = k + (α + mq + hmq) α + β (α + nq + lnq) α + β hjk(q + gq + gq + lq + mq) deghlq dghjlq dghklq dghlpq eghjlq jk(α + β) d(γ + dghlpq + ghilpq) ijk(α + β) e(γ + eghloq + ghkloq) ijk(α + β) ijl(q + gq + hq + nq + gq) deghlq dghjlq eghilq eghjlq eghloq ij(α + β) (α + mq + hmq)(α + nq + lnq) mn(α + β) h(α + nq + lnq) α + β γ + dghlpq + ghilpq jk(α + β) (ijk(α + β) + γ + dghlpq + ghilpq)(ijk(α + β) + γ + eghloq + ghkloq) ik(α + β)(ijk(α + β) + γ + δ) k(γ + eghloq + ghkloq) ijk(α + β) 26

27 l(α + mq + hmq) L = α + β M = N = O = P = Q = References α + mq + hmq gn α + nq + lnq gm o(ijk(α + β) + γ + dghlpq + ghilpq) ijk(α + β) + γ + δ p(ijk(α + β) + γ + eghloq + ghkloq) ijk(α + β) + γ + δ hgjlopq ijk(α + β) + γ + δ [1] E.B. Curtis, D. Ingermn, nd J.A. Morrow. Circulr Plnr Grphs nd Resistor Networks. In: Liner Alger nd its Applictions 283 (1998), pp [2] E.B. Curtis, E. Mooers, nd J.A. Morrow. Finding the Conductors in Circulr Networks from Boundry Mesurments. In: Liner Alger nd its Applictions 28 (1994), pp [3] Y.C. de Verdiere, I. Gitler, nd D. Vertign. Reseux electriques plnires II. In: Comment. Mth. Helv (1996), pp [4] R. Kenyon. The Lplcin on plnr grphs nd grphs on surfces. In: ArXiv e-prints (Mr. 2012). rxiv: [mth.pr]. [5] R. Kenyon nd D. Wilson. Boundry prtitions in trees nd dimers. In: Trns. Amer. Mth. Soc (2011), [6] R. Kenyon nd D. Wilson. Comintorics of triprtite oundry connections for trees nd dimers. In: Electron. J. Comin (2009). [7] T. Lm nd P. Pylyvskyy. Inverse prolem in cylindricl electricl networks. In: ArXiv e-prints (Apr. 2011). rxiv: [mth.co]. 27