Greedoid polynomil, hip-firing, nd G-prking funtion for direted grphs Swee Hong Chn Cornell University Connetions in Disrete Mthemtis June 15, 2015
Motivtion Tutte polynomil [Tut54] is polynomil defined for undireted grphs tht hs pplitions to severl rnhes of mthemtis (grph oloring, knot theory, Ising model, et.). We re interested in generliztion of Tutte polynomil for direted grphs nd its onnetions to other topis in disrete mthemtis. Known Tutte-like polynomil for direted grphs: Cover polynomil [CG95], Greedoid polynomil [BKL85]. In this tlk, G is direted grph tht my hve loops nd multiple edges, nd is strongly onneted.
Motivtion Tutte polynomil [Tut54] is polynomil defined for undireted grphs tht hs pplitions to severl rnhes of mthemtis (grph oloring, knot theory, Ising model, et.). We re interested in generliztion of Tutte polynomil for direted grphs nd its onnetions to other topis in disrete mthemtis. Known Tutte-like polynomil for direted grphs: Cover polynomil [CG95], Greedoid polynomil [BKL85]. In this tlk, G is direted grph tht my hve loops nd multiple edges, nd is strongly onneted.
Motivtion Tutte polynomil [Tut54] is polynomil defined for undireted grphs tht hs pplitions to severl rnhes of mthemtis (grph oloring, knot theory, Ising model, et.). We re interested in generliztion of Tutte polynomil for direted grphs nd its onnetions to other topis in disrete mthemtis. Known Tutte-like polynomil for direted grphs: Cover polynomil [CG95], Greedoid polynomil [BKL85]. In this tlk, G is direted grph tht my hve loops nd multiple edges, nd is strongly onneted.
Motivtion Tutte polynomil [Tut54] is polynomil defined for undireted grphs tht hs pplitions to severl rnhes of mthemtis (grph oloring, knot theory, Ising model, et.). We re interested in generliztion of Tutte polynomil for direted grphs nd its onnetions to other topis in disrete mthemtis. Known Tutte-like polynomil for direted grphs: Cover polynomil [CG95], Greedoid polynomil [BKL85]. In this tlk, G is direted grph tht my hve loops nd multiple edges, nd is strongly onneted.
Rooted spnning tree Choose root vertex V (G). In -rooted spnning tree, there is uniue direted pth from to v for eh vertex v V (G). -rooted spnning trees
Externl tivity Fix totl order < e on E(G). Let T e -rooted spnning tree, nd let e E(G) \ E(T ). The grph T {e} hs uniue iruit C.
Externl tivity Fix totl order < e on E(G). Let T e -rooted spnning tree, nd let e E(G) \ E(T ). The grph T {e} hs uniue iruit C. e
Externl tivity Fix totl order < e on E(G). Let T e -rooted spnning tree, nd let e E(G) \ E(T ). The grph T {e} hs uniue iruit C. e
Externl tivity Fix totl order < e on E(G). Let T e -rooted spnning tree, nd let e E(G) \ E(T ). The grph T {e} hs uniue iruit C. e The iruit C deomposes to two direted pths P 1 nd P 2. Let P 1 e the pth tht ontins e. P 1 P 2
Externl tivity Fix totl order < e on E(G). Let T e -rooted spnning tree, nd let e E(G) \ E(T ). The grph T {e} hs uniue iruit C. e The iruit C deomposes to two direted pths P 1 nd P 2. Let P 1 e the pth tht ontins e. P 1 P 2 The edge e is externlly tive w.r.t.t if the smllest edge of E(C) is ontined in E(P 1 ).
Greedoid polynomil The externl tivity of T, denoted y ext(t ), is the numer of edges tht re externlly tive w.r.t T. The greedoid polynomil of G w.r.t root vertex is the polynomil λ (G; y) = T y ext(t ). The polynomil λ (G; y) does not depend on the totl ordering < e. If G is undireted, then λ (G; y) = y Ē(G) T (G; 1, y).
Greedoid polynomil The externl tivity of T, denoted y ext(t ), is the numer of edges tht re externlly tive w.r.t T. The greedoid polynomil of G w.r.t root vertex is the polynomil λ (G; y) = T y ext(t ). The polynomil λ (G; y) does not depend on the totl ordering < e. If G is undireted, then λ (G; y) = y Ē(G) T (G; 1, y).
Greedoid polynomil (td.) Things tht re different from Tutte polynomil: There is no notion of internl tivity. It stisfies wek deletion-ontrtion reurrene, i.e. if e is n edge rooted t, then: { yλ (G \ {e}; y) if e is loop; λ (G; y) = λ (G \ {e}; y) + λ (G/{e}; y) otherwise. The polynomil λ (G; y) depends on the hoie of root.
G-prking funtion A funtion f : V (G) \ {} N is G-prking funtion w.r.t. [PS04] if for ll nonempty A V (G) \ {}, there is v A so tht f (v) < numer of edges from V (G) \ A to v. Exmple. 20 1 2 5 31 4 3 40 The level of f, denoted y lvl(f ), is lvl(f ) = E(G) V (G) + 1 f (v). v V (G)\{}
G-prking funtion A funtion f : V (G) \ {} N is G-prking funtion w.r.t. [PS04] if for ll nonempty A V (G) \ {}, there is v A so tht f (v) < numer of edges from V (G) \ A to v. Exmple. 20 1 2 5 31 4 3 40 The level of f, denoted y lvl(f ), is lvl(f ) = E(G) V (G) + 1 f (v). v V (G)\{}
G-prking funtion A funtion f : V (G) \ {} N is G-prking funtion w.r.t. [PS04] if for ll nonempty A V (G) \ {}, there is v A so tht f (v) < numer of edges from V (G) \ A to v. Exmple. 20 1 2 5 31 4 3 40 The level of f, denoted y lvl(f ), is lvl(f ) = E(G) V (G) + 1 f (v). v V (G)\{}
Connetions to greedoid polynomil Theorem 1 (C.,2015) y lvl(f ) = f T y ext(t ) = λ (G; y). Corollry 2 There is ijetion etween G-prking funtions w.r.t nd -rooted spnning trees of G tht trnsltes level to externl tivity. Corollry 2 is generliztion of Cori-Le Borgne ijetion for undireted grphs [CLB03].
Connetions to greedoid polynomil Theorem 1 (C.,2015) y lvl(f ) = f T y ext(t ) = λ (G; y). Corollry 2 There is ijetion etween G-prking funtions w.r.t nd -rooted spnning trees of G tht trnsltes level to externl tivity. Corollry 2 is generliztion of Cori-Le Borgne ijetion for undireted grphs [CLB03].
Connetions to greedoid polynomil Theorem 1 (C.,2015) y lvl(f ) = f T y ext(t ) = λ (G; y). Corollry 2 There is ijetion etween G-prking funtions w.r.t nd -rooted spnning trees of G tht trnsltes level to externl tivity. Corollry 2 is generliztion of Cori-Le Borgne ijetion for undireted grphs [CLB03].
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S = {}, E = {1, 2, 3, 4, 5}, E S = {1, 5}. 20 31 40
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S = {}, E = {1, 2, 3, 4, 5}, E S = {1,5}. 20 5 31 40
The ijetion The grph G nd the G-prking funtion. 5 1 2 3 4 3 2 4 1 0 0 The lgorithm. S = {}, E = {1, 2, 3, 4, 5}, E S = {1,5}. 3 2 4 5 0 0 0
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S = {}, E = {1, 2, 3, 4}, E S ={1}. 20 30 40
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S = {}, E = { 1, 2, 3, 4}, E S ={1}. 1 1 0 0
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S={, }, E = { 1, 2, 3, 4}, E S ={1}. 1 1 1 0 0
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S={, }, E = {2, 3, 4}, E S ={2}. 1 1 0 0
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S={,, }, E = { 2, 3, 4}, E S ={2}. 1 1 2 2 2 1 0
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S={,, }, E = {3, 4}, E S ={3}. 1 1 2 1 0
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S={,,, }, E = { 3, 4}, E S ={3}. 1 1 2 1 3 2 1 3
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S={,,, }, E = {4}, E S ={4}. 1 1 2 1 3 1
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S={,,, }, E = { 4}, E S ={4}. 1 1 2 1 3 4 1
The ijetion The grph G nd the G-prking funtion. 20 1 2 5 31 4 3 40 The lgorithm. S={,,, }, E = {}, E S ={}. Algorithm stops. 1 1 2 1 3 1
Chip-firing gme The sinkless Chip-firing gme [BTW88], lso known s elin sndpile model nd Bk-Tng-Wiesenfield model, is dynmil system on digrph G. Eh vertex v V (G) is ssigned (v) hips, nd v is unstle if (v) is greter thn or eul to the outdegree of v. If v is unstle, then we n topple the vertex y sending hip through eh of its outgoing edges to its neighors. Exmple of toppling moves.
Chip-firing gme The sinkless Chip-firing gme [BTW88], lso known s elin sndpile model nd Bk-Tng-Wiesenfield model, is dynmil system on digrph G. Eh vertex v V (G) is ssigned (v) hips, nd v is unstle if (v) is greter thn or eul to the outdegree of v. If v is unstle, then we n topple the vertex y sending hip through eh of its outgoing edges to its neighors. Exmple of toppling moves. 20 12 30 40
Chip-firing gme The sinkless Chip-firing gme [BTW88], lso known s elin sndpile model nd Bk-Tng-Wiesenfield model, is dynmil system on digrph G. Eh vertex v V (G) is ssigned (v) hips, nd v is unstle if (v) is greter thn or eul to the outdegree of v. If v is unstle, then we n topple the vertex y sending hip through eh of its outgoing edges to its neighors. Exmple of toppling moves. 21 10 31 40
Chip-firing gme The sinkless Chip-firing gme [BTW88], lso known s elin sndpile model nd Bk-Tng-Wiesenfield model, is dynmil system on digrph G. Eh vertex v V (G) is ssigned (v) hips, nd v is unstle if (v) is greter thn or eul to the outdegree of v. If v is unstle, then we n topple the vertex y sending hip through eh of its outgoing edges to its neighors. Exmple of toppling moves. 20 10 32 40
Chip-firing gme The sinkless Chip-firing gme [BTW88], lso known s elin sndpile model nd Bk-Tng-Wiesenfield model, is dynmil system on digrph G. Eh vertex v V (G) is ssigned (v) hips, nd v is unstle if (v) is greter thn or eul to the outdegree of v. If v is unstle, then we n topple the vertex y sending hip through eh of its outgoing edges to its neighors. Exmple of toppling moves. 20 10 31 41
Reurrent onfigurtion A hip onfigurtion is funtion : V (G) N, where (v) is the numer of hips in vertex v. We write if there is seuene of toppling moves from to. A hip onfigurtion is reurrent if: is unstle; If, then. For two reurrent onfigurtions nd, we write if nd. The level of reurrent euivlene lss [] is lvl([]) = v V (G) (v). We use r n to denote the numer of reurrent euivlene lsses with level n.
Reurrent onfigurtion A hip onfigurtion is funtion : V (G) N, where (v) is the numer of hips in vertex v. We write if there is seuene of toppling moves from to. A hip onfigurtion is reurrent if: is unstle; If, then. For two reurrent onfigurtions nd, we write if nd. The level of reurrent euivlene lss [] is lvl([]) = v V (G) (v). We use r n to denote the numer of reurrent euivlene lsses with level n.
Reurrent onfigurtion A hip onfigurtion is funtion : V (G) N, where (v) is the numer of hips in vertex v. We write if there is seuene of toppling moves from to. A hip onfigurtion is reurrent if: is unstle; If, then. For two reurrent onfigurtions nd, we write if nd. The level of reurrent euivlene lss [] is lvl([]) = v V (G) (v). We use r n to denote the numer of reurrent euivlene lsses with level n.
Connetions to greedoid polynomil Theorem 3 (C.,2015) If G is n Eulerin digrph, then n 0 r n y n = λ (G; y) (1 y). Theorem 4 (C., 2015) The greedoid polynomil λ (G; y) does not depend on the hoie of root vertex if nd only if G is n Eulerin digrph. Theorem 5 (C., 2015) Let G e the reverse of the digrph G. We hve G is n Eulerin digrph if nd only if λ (G; y) = λ (G ; y) for ll V (G).
Connetions to greedoid polynomil Theorem 3 (C.,2015) If G is n Eulerin digrph, then n 0 r n y n = λ (G; y) (1 y). Theorem 4 (C., 2015) The greedoid polynomil λ (G; y) does not depend on the hoie of root vertex if nd only if G is n Eulerin digrph. Theorem 5 (C., 2015) Let G e the reverse of the digrph G. We hve G is n Eulerin digrph if nd only if λ (G; y) = λ (G ; y) for ll V (G).
Connetions to greedoid polynomil Theorem 3 (C.,2015) If G is n Eulerin digrph, then n 0 r n y n = λ (G; y) (1 y). Theorem 4 (C., 2015) The greedoid polynomil λ (G; y) does not depend on the hoie of root vertex if nd only if G is n Eulerin digrph. Theorem 5 (C., 2015) Let G e the reverse of the digrph G. We hve G is n Eulerin digrph if nd only if λ (G; y) = λ (G ; y) for ll V (G).
Further reserh Find ijetive proof for Theorem 4 nd Theorem 5. Prove or disprove tht λ (G; y) is unimodl.
Questions?
Thnk you!
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