Minimally non-balanced diamond-free graphs Anna Galluccio Istituto - - PowerPoint PPT Presentation

Minimally non-balanced diamond-free graphs

Anna Galluccio

Istituto Analisi Sistemi ed Informatica (IASI-CNR)- Italy

JOINT WORK WITH: N. Apollonio, Istituto delle Applicazioni del Calcolo (IAC-CNR)

Aussois Workshop January 2015

K¨

• nig property

Given a 0, 1-matrix A ν(A) =max # of pairwise non-intersecting rows (matching) τ(A) =min # of columns intersecting all rows (transversal) A satisfies the K¨

• nig property if and only if ν(A) = τ(A).

C2k+1 =           1 1 · · · 1 1 · · · 1 ... . . . . . . . . . ... 1 1 1 · · · 1          

• dd cycle matrix has

ν(C2k+1) = k < τ(C2k+1) = k + 1

K¨

• nig property

Given a 0, 1-matrix A ν(A) =max # of pairwise non-intersecting rows (matching) τ(A) =min # of columns intersecting all rows (transversal) A satisfies the K¨

• nig property if and only if ν(A) = τ(A).

C2k+1 =           1 1 · · · 1 1 · · · 1 ... . . . . . . . . . ... 1 1 1 · · · 1          

• dd cycle matrix has

ν(C2k+1) = k < τ(C2k+1) = k + 1

K¨

• nig property

Given a 0, 1-matrix A ν(A) =max # of pairwise non-intersecting rows (matching) τ(A) =min # of columns intersecting all rows (transversal) A satisfies the K¨

• nig property if and only if ν(A) = τ(A).

C2k+1 =           1 1 · · · 1 1 · · · 1 ... . . . . . . . . . ... 1 1 1 · · · 1          

• dd cycle matrix has

ν(C2k+1) = k < τ(C2k+1) = k + 1

K¨

• nig property

Given a 0, 1-matrix A ν(A) =max # of pairwise non-intersecting rows (matching) τ(A) =min # of columns intersecting all rows (transversal) A satisfies the K¨

• nig property if and only if ν(A) = τ(A).

C2k+1 =           1 1 · · · 1 1 · · · 1 ... . . . . . . . . . ... 1 1 1 · · · 1          

• dd cycle matrix has

ν(C2k+1) = k < τ(C2k+1) = k + 1

Balanced matrices and balanced graphs

• Def. A is balanced iff ν(A′) = τ(A′) for any A′ submatrix of A.

Theorem (Berge, Las Vergnas 1972) A balanced if and only if A ⊇ C2k+1, k ≥ 1, as a submatrix G graph, AG denote clique-matrix of G (matrix whose rows are the incidence vectors of maximal cliques of G)

• Def. G is balanced if and only if AG is balanced
• Def. G minimally non-balanced (MNB) if and only G is not

balanced but each its proper induced subgraphs is balanced.

Balanced matrices and balanced graphs

• Def. A is balanced iff ν(A′) = τ(A′) for any A′ submatrix of A.

Theorem (Berge, Las Vergnas 1972) A balanced if and only if A ⊇ C2k+1, k ≥ 1, as a submatrix G graph, AG denote clique-matrix of G (matrix whose rows are the incidence vectors of maximal cliques of G)

• Def. G is balanced if and only if AG is balanced
• Def. G minimally non-balanced (MNB) if and only G is not

balanced but each its proper induced subgraphs is balanced.

Balanced matrices and balanced graphs

• Def. A is balanced iff ν(A′) = τ(A′) for any A′ submatrix of A.

Theorem (Berge, Las Vergnas 1972) A balanced if and only if A ⊇ C2k+1, k ≥ 1, as a submatrix G graph, AG denote clique-matrix of G (matrix whose rows are the incidence vectors of maximal cliques of G)

• Def. G is balanced if and only if AG is balanced
• Def. G minimally non-balanced (MNB) if and only G is not

balanced but each its proper induced subgraphs is balanced.

Balanced matrices and balanced graphs

• Def. A is balanced iff ν(A′) = τ(A′) for any A′ submatrix of A.

Theorem (Berge, Las Vergnas 1972) A balanced if and only if A ⊇ C2k+1, k ≥ 1, as a submatrix G graph, AG denote clique-matrix of G (matrix whose rows are the incidence vectors of maximal cliques of G)

• Def. G is balanced if and only if AG is balanced
• Def. G minimally non-balanced (MNB) if and only G is not

balanced but each its proper induced subgraphs is balanced.

Minimally non-balanced graphs

It is natural to ask for characterization of minimally non-balanced graphs but the problem does not appear to be easy. first attempt to attack the general problem is due to Bonomo, Duran, Lin and Swarzficter (2002) with extended

• dd sun

minimal characterizations exist for special classes of graphs: chordal, line graphs, paw free, complement of line graphs [Bonomo,Chudnovsky,Duran, Safe,Wagler,..] decomposition theorem of Conforti, Cornuejols and Rao (1999) yields a polynomial-time recognition algorithm more recently, a connection with the study of minimally clique-imperfect graphs.

Minimally non-balanced graphs

It is natural to ask for characterization of minimally non-balanced graphs but the problem does not appear to be easy. first attempt to attack the general problem is due to Bonomo, Duran, Lin and Swarzficter (2002) with extended

• dd sun

minimal characterizations exist for special classes of graphs: chordal, line graphs, paw free, complement of line graphs [Bonomo,Chudnovsky,Duran, Safe,Wagler,..] decomposition theorem of Conforti, Cornuejols and Rao (1999) yields a polynomial-time recognition algorithm more recently, a connection with the study of minimally clique-imperfect graphs.

Minimally non-balanced graphs

It is natural to ask for characterization of minimally non-balanced graphs but the problem does not appear to be easy. first attempt to attack the general problem is due to Bonomo, Duran, Lin and Swarzficter (2002) with extended

• dd sun

minimal characterizations exist for special classes of graphs: chordal, line graphs, paw free, complement of line graphs [Bonomo,Chudnovsky,Duran, Safe,Wagler,..] decomposition theorem of Conforti, Cornuejols and Rao (1999) yields a polynomial-time recognition algorithm more recently, a connection with the study of minimally clique-imperfect graphs.

Clique-perfection

τc(G) = min # of vertices that meets all the maximal cliques of G. αc(G) = max # of pairwise vertex-disjoint cliques of G. A graph G is clique-perfect if τc(G) = αc(G) for each induced subgraph G′ of G. These graph-invariants were already studied by Tuza and Lehel during ’80s, but recently Bonomo, Chudnovsky and Duran (2007) explicitly asked for complexity of recognizing diamond-free clique-perfect graphs, characterization of minimally clique-imperfect diamond-free graphs.

Clique-perfection

τc(G) = min # of vertices that meets all the maximal cliques of G. αc(G) = max # of pairwise vertex-disjoint cliques of G. A graph G is clique-perfect if τc(G) = αc(G) for each induced subgraph G′ of G. These graph-invariants were already studied by Tuza and Lehel during ’80s, but recently Bonomo, Chudnovsky and Duran (2007) explicitly asked for complexity of recognizing diamond-free clique-perfect graphs, characterization of minimally clique-imperfect diamond-free graphs.

Clique-perfection

τc(G) = min # of vertices that meets all the maximal cliques of G. αc(G) = max # of pairwise vertex-disjoint cliques of G. A graph G is clique-perfect if τc(G) = αc(G) for each induced subgraph G′ of G. These graph-invariants were already studied by Tuza and Lehel during ’80s, but recently Bonomo, Chudnovsky and Duran (2007) explicitly asked for complexity of recognizing diamond-free clique-perfect graphs, characterization of minimally clique-imperfect diamond-free graphs.

Clique-perfection

τc(G) = min # of vertices that meets all the maximal cliques of G. αc(G) = max # of pairwise vertex-disjoint cliques of G. A graph G is clique-perfect if τc(G) = αc(G) for each induced subgraph G′ of G. These graph-invariants were already studied by Tuza and Lehel during ’80s, but recently Bonomo, Chudnovsky and Duran (2007) explicitly asked for complexity of recognizing diamond-free clique-perfect graphs, characterization of minimally clique-imperfect diamond-free graphs.

Clique-perfection

τc(G) = min # of vertices that meets all the maximal cliques of G. αc(G) = max # of pairwise vertex-disjoint cliques of G. A graph G is clique-perfect if τc(G) = αc(G) for each induced subgraph G′ of G. These graph-invariants were already studied by Tuza and Lehel during ’80s, but recently Bonomo, Chudnovsky and Duran (2007) explicitly asked for complexity of recognizing diamond-free clique-perfect graphs, characterization of minimally clique-imperfect diamond-free graphs.

Clique-perfection

τc(G) = min # of vertices that meets all the maximal cliques of G. αc(G) = max # of pairwise vertex-disjoint cliques of G. A graph G is clique-perfect if τc(G) = αc(G) for each induced subgraph G′ of G. These graph-invariants were already studied by Tuza and Lehel during ’80s, but recently Bonomo, Chudnovsky and Duran (2007) explicitly asked for complexity of recognizing diamond-free clique-perfect graphs, characterization of minimally clique-imperfect diamond-free graphs.

Clique-perfection versus balancedness

K3 K4 K5 K7 K0 K1 K2 K3 K4 K5 K6 K7 K8 K9 K1 K2 K6

ν(AG) = ν(C9) = 4 < τ(AG) = 5 = τ(C9) ν(AG) = τ(AG) = 3

Theorem (G. , Apollonio 2013, Safe P.h.D.Thesis 2009) G diamond-free. Then G clique-perfect ⇔ G balanced

Clique-perfection versus balancedness

K3 K4 K5 K7 K0 K1 K2 K3 K4 K5 K6 K7 K8 K9 K1 K2 K6

ν(AG) = ν(C9) = 4 < τ(AG) = 5 = τ(C9) ν(AG) = τ(AG) = 3

Theorem (G. , Apollonio 2013, Safe P.h.D.Thesis 2009) G diamond-free. Then G clique-perfect ⇔ G balanced

Non-balanced diamond-free graphs

G diamond-free graph if and only if AG is linear (i.e., AG does not contain 1 1

1 1

• as a submatrix).

g(A)= min order of an odd cycle submatrix of A B↑ (up-matrix) obtained from B by removing its dominated rows Lemma Let G be diamond-free non-balanced. If g(AG) = n then AG contains an up-matrix Cn

F

• such that

either F = ∅

• r the rows of F have ≥ 3

nonzero entries that correspond to stable sets

• n the odd cycle C

(multisun)

Non-balanced diamond-free graphs

G diamond-free graph if and only if AG is linear (i.e., AG does not contain 1 1

1 1

• as a submatrix).

g(A)= min order of an odd cycle submatrix of A B↑ (up-matrix) obtained from B by removing its dominated rows Lemma Let G be diamond-free non-balanced. If g(AG) = n then AG contains an up-matrix Cn

F

• such that

either F = ∅

• r the rows of F have ≥ 3

nonzero entries that correspond to stable sets

• n the odd cycle C

(multisun)

Non-balanced diamond-free graphs

G diamond-free graph if and only if AG is linear (i.e., AG does not contain 1 1

1 1

• as a submatrix).

g(A)= min order of an odd cycle submatrix of A B↑ (up-matrix) obtained from B by removing its dominated rows Lemma Let G be diamond-free non-balanced. If g(AG) = n then AG contains an up-matrix Cn

F

• such that

either F = ∅

• r the rows of F have ≥ 3

nonzero entries that correspond to stable sets

• n the odd cycle C

(multisun)

Non-balanced diamond-free graphs

G diamond-free graph if and only if AG is linear (i.e., AG does not contain 1 1

1 1

• as a submatrix).

g(A)= min order of an odd cycle submatrix of A B↑ (up-matrix) obtained from B by removing its dominated rows Lemma Let G be diamond-free non-balanced. If g(AG) = n then AG contains an up-matrix Cn

F

• such that

either F = ∅

• r the rows of F have ≥ 3

nonzero entries that correspond to stable sets

• n the odd cycle C

(multisun)

Theorem G diamond-free with n vertices, n > 3. Then g(AG) = n ⇐ ⇒ either AG = Cn

• r AG =

Cn

F

• and for each

F ′ ⊆ F, Cn

F ′

• does not contain

Cg, g < n, as up-matrix Sub-multisun is obtained from a multisun by deleting the edges of some inscribed clique Definition G multisun is Hereditarily-Odd-Hole-free (HOH-free) iff every sub-multisun of G is odd-hole-free

Theorem G diamond-free with n vertices, n > 3. Then g(AG) = n ⇐ ⇒ either AG = Cn

• r AG =

Cn

F

• and for each

F ′ ⊆ F, Cn

F ′

• does not contain

Cg, g < n, as up-matrix Sub-multisun is obtained from a multisun by deleting the edges of some inscribed clique Definition G multisun is Hereditarily-Odd-Hole-free (HOH-free) iff every sub-multisun of G is odd-hole-free

Theorem G diamond-free with n vertices, n > 3. Then g(AG) = n ⇐ ⇒ either AG = Cn

• r AG =

Cn

F

• and for each

F ′ ⊆ F, Cn

F ′

• does not contain

Cg, g < n, as up-matrix Sub-multisun is obtained from a multisun by deleting the edges of some inscribed clique Definition G multisun is Hereditarily-Odd-Hole-free (HOH-free) iff every sub-multisun of G is odd-hole-free

Theorem G diamond-free with n vertices, n > 3. Then g(AG) = n ⇐ ⇒ either AG = Cn

• r AG =

Cn

F

• and for each

F ′ ⊆ F, Cn

F ′

• does not contain

Cg, g < n, as up-matrix Sub-multisun is obtained from a multisun by deleting the edges of some inscribed clique Definition G multisun is Hereditarily-Odd-Hole-free (HOH-free) iff every sub-multisun of G is odd-hole-free

Minimally non-balanced graphs

G diamond-free MNB iff G is either an odd hole or a Hereditary Odd Hole Free (HOH-free) multisun

Minimally non-balanced graphs

G diamond-free MNB iff G is either an odd hole or a Hereditary Odd Hole Free (HOH-free) multisun

N-conditions

The inscribed cliques intersect in the same vertex and are otherwise vertex disjoint Odd between two cliques Even within vertices of the same cliques whose interior is not in another clique even Cliques are

• dd

Canonical labeling

Theorem If G is HOH-free multisun, the G satisfies the N-conditions. Canonical Labeling Let G be a multisun with rim C. Set Σ = {ǫ, σ} ∪ {x | X is an inscribed clique of G}. Let f : V(C) → Σ be defined as follows: f(v) =    σ if v belongs to more than one inscribed clique of G; x if v belongs to the inscribed clique X of G; ǫ if v belongs to no inscribed clique of G. Σ − {ǫ, σ} are called the proper letters.

Canonical labeling

Theorem If G is HOH-free multisun, the G satisfies the N-conditions. Canonical Labeling Let G be a multisun with rim C. Set Σ = {ǫ, σ} ∪ {x | X is an inscribed clique of G}. Let f : V(C) → Σ be defined as follows: f(v) =    σ if v belongs to more than one inscribed clique of G; x if v belongs to the inscribed clique X of G; ǫ if v belongs to no inscribed clique of G. Σ − {ǫ, σ} are called the proper letters.

aǫ2aǫ4aǫ2aǫ6aǫ8aǫ2aǫ20 σǫ4aǫ2aǫ6aǫbǫ2bǫ3aǫ2aǫbǫ3cǫ2cǫ3bǫ2bǫ2bǫaǫ2

a a a a a a a ǫ4 ǫ2 ǫ6 ǫ2 ǫ2 σ a a a b b a a b c c b b b a ǫ ǫ4 ǫ2 ǫ6 ǫ2 ǫ2 ǫ2 ǫ ǫ3 ǫ2 ǫ3 ǫ ǫ2 ǫ3 ǫ2 ǫ8 ǫ20 (b) (a)

• Being a multisun that satisfies N-conditions is preserved

under taking even subdivision/ contraction of the rim. Two words are pattern-equivalent if one can be transformed into the other by repeatedly applying one of the following

• perations:

shifting the indices, reversing the order of reading, replace the interval ǫǫ by the empty word φ (even contraction), replace φ (empty) by k times ǫǫ with k ≥ 1 (even subdivision),

• Being a multisun that satisfies N-conditions is preserved

under taking even subdivision/ contraction of the rim. Two words are pattern-equivalent if one can be transformed into the other by repeatedly applying one of the following

• perations:

shifting the indices, reversing the order of reading, replace the interval ǫǫ by the empty word φ (even contraction), replace φ (empty) by k times ǫǫ with k ≥ 1 (even subdivision),

s-word

⋄ aǫ2aǫ4aǫ2aǫ6aǫ8aǫ2aǫ20 → a7 ⋄ σǫ4aǫ2aǫ6aǫbǫ2bǫ3aǫ2aǫbǫ3cǫ2cǫ3bǫ2bǫ2bǫaǫ2 → σa3ǫb2ǫa2ǫbǫc2ǫb3ǫa A class of pattern-equivalent words

called s-word and it has one of the following forms: i) if Σ = {ǫ, a}, then

ii) if Σ = {σ, ǫ, a, b, . . .}, then

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is ] with

λ1, . . . , λs positive integers and

• xih = xih+1 h = 1, 2, . . . , s − 1 and
• the sum of the exponents of each proper letter is even.

s-word

⋄ aǫ2aǫ4aǫ2aǫ6aǫ8aǫ2aǫ20 → a7 ⋄ σǫ4aǫ2aǫ6aǫbǫ2bǫ3aǫ2aǫbǫ3cǫ2cǫ3bǫ2bǫ2bǫaǫ2 → σa3ǫb2ǫa2ǫbǫc2ǫb3ǫa A class of pattern-equivalent words

called s-word and it has one of the following forms: i) if Σ = {ǫ, a}, then

ii) if Σ = {σ, ǫ, a, b, . . .}, then

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is ] with

λ1, . . . , λs positive integers and

• xih = xih+1 h = 1, 2, . . . , s − 1 and
• the sum of the exponents of each proper letter is even.

s-word

⋄ aǫ2aǫ4aǫ2aǫ6aǫ8aǫ2aǫ20 → a7 ⋄ σǫ4aǫ2aǫ6aǫbǫ2bǫ3aǫ2aǫbǫ3cǫ2cǫ3bǫ2bǫ2bǫaǫ2 → σa3ǫb2ǫa2ǫbǫc2ǫb3ǫa A class of pattern-equivalent words

called s-word and it has one of the following forms: i) if Σ = {ǫ, a}, then

ii) if Σ = {σ, ǫ, a, b, . . .}, then

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is ] with

λ1, . . . , λs positive integers and

• xih = xih+1 h = 1, 2, . . . , s − 1 and
• the sum of the exponents of each proper letter is even.

s-word

⋄ aǫ2aǫ4aǫ2aǫ6aǫ8aǫ2aǫ20 → a7 ⋄ σǫ4aǫ2aǫ6aǫbǫ2bǫ3aǫ2aǫbǫ3cǫ2cǫ3bǫ2bǫ2bǫaǫ2 → σa3ǫb2ǫa2ǫbǫc2ǫb3ǫa A class of pattern-equivalent words

called s-word and it has one of the following forms: i) if Σ = {ǫ, a}, then

ii) if Σ = {σ, ǫ, a, b, . . .}, then

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is ] with

λ1, . . . , λs positive integers and

• xih = xih+1 h = 1, 2, . . . , s − 1 and
• the sum of the exponents of each proper letter is even.

s-word

⋄ aǫ2aǫ4aǫ2aǫ6aǫ8aǫ2aǫ20 → a7 ⋄ σǫ4aǫ2aǫ6aǫbǫ2bǫ3aǫ2aǫbǫ3cǫ2cǫ3bǫ2bǫ2bǫaǫ2 → σa3ǫb2ǫa2ǫbǫc2ǫb3ǫa A class of pattern-equivalent words

called s-word and it has one of the following forms: i) if Σ = {ǫ, a}, then

ii) if Σ = {σ, ǫ, a, b, . . .}, then

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is ] with

λ1, . . . , λs positive integers and

• xih = xih+1 h = 1, 2, . . . , s − 1 and
• the sum of the exponents of each proper letter is even.

Sunoids and sunwords

A sunoid is a multisun G that hereditarily satisfies the N-conditions, i.e., each sub-multisun of G satisfies the N-conditions. The class of sunoids is denoted by S∗. If S∗∗ denotes the class of HOH-free multisuns and S denotes the class of multisuns satisfying the N-conditions, then we clearly have S∗∗ ⊆ S∗ ⊆ S. Aim S∗∗ = S∗. Call sunword the s-word of a sunoid Characterizing sunoids among multisuns satisfying the N-conditions is the same as characterizing sunwords among s-words.

Sunoids and sunwords

A sunoid is a multisun G that hereditarily satisfies the N-conditions, i.e., each sub-multisun of G satisfies the N-conditions. The class of sunoids is denoted by S∗. If S∗∗ denotes the class of HOH-free multisuns and S denotes the class of multisuns satisfying the N-conditions, then we clearly have S∗∗ ⊆ S∗ ⊆ S. Aim S∗∗ = S∗. Call sunword the s-word of a sunoid Characterizing sunoids among multisuns satisfying the N-conditions is the same as characterizing sunwords among s-words.

Sunoids and sunwords

A sunoid is a multisun G that hereditarily satisfies the N-conditions, i.e., each sub-multisun of G satisfies the N-conditions. The class of sunoids is denoted by S∗. If S∗∗ denotes the class of HOH-free multisuns and S denotes the class of multisuns satisfying the N-conditions, then we clearly have S∗∗ ⊆ S∗ ⊆ S. Aim S∗∗ = S∗. Call sunword the s-word of a sunoid Characterizing sunoids among multisuns satisfying the N-conditions is the same as characterizing sunwords among s-words.

Sunoids and sunwords

A sunoid is a multisun G that hereditarily satisfies the N-conditions, i.e., each sub-multisun of G satisfies the N-conditions. The class of sunoids is denoted by S∗. If S∗∗ denotes the class of HOH-free multisuns and S denotes the class of multisuns satisfying the N-conditions, then we clearly have S∗∗ ⊆ S∗ ⊆ S. Aim S∗∗ = S∗. Call sunword the s-word of a sunoid Characterizing sunoids among multisuns satisfying the N-conditions is the same as characterizing sunwords among s-words.

Parity conditions

Two necessary and sufficient conditions:

• first show that sunwords satisfy certain parity conditions on

the exponents of the proper letters

• then show that sunwords satisfy a sort of continuity

property with respect to a linear order on the proper letters, Lemma If

σxλ1

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is that satisfies the following parity

conditions: (i) λ1 and λs are both odd; (ii) λh is odd if and only if xih−1 = xih+1, h = 2, . . . , s − 1.

Parity conditions

Two necessary and sufficient conditions:

• first show that sunwords satisfy certain parity conditions on

the exponents of the proper letters

• then show that sunwords satisfy a sort of continuity

property with respect to a linear order on the proper letters, Lemma If

σxλ1

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is that satisfies the following parity

conditions: (i) λ1 and λs are both odd; (ii) λh is odd if and only if xih−1 = xih+1, h = 2, . . . , s − 1.

a7 σa3ǫb2ǫa2ǫbǫc2ǫb3ǫa

a a a a a a a ǫ4 ǫ2 ǫ6 ǫ2 ǫ2 σ a a a b b a a b c c b b b a ǫ ǫ4 ǫ2 ǫ6 ǫ2 ǫ2 ǫ2 ǫ ǫ3 ǫ2 ǫ3 ǫ ǫ2 ǫ3 ǫ2 ǫ8 ǫ20 (b) (a)

Jump-freeness

An s-word induces a finite chain σ ≺ a ≺ b ≺ c · · · . Given

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is ] s-word with xi0 = σ and

linear order σ ≺ a ≺ b · · · .

• two letters x and y form a cover pair if x y and

∃z = x, y such that x ≺ z ≺ y.

• two letters xih and xih+1 , h ≥ 0 are a jump on x and y in

if x and y is not a cover pair, and either xih = x, xih+1 = y or xih = y, xih+1 = x (sums mod s + 1). An s-word is jump-free if it contains no jump for any {x, y} = ǫ. Theorem If

Jump-freeness

An s-word induces a finite chain σ ≺ a ≺ b ≺ c · · · . Given

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is ] s-word with xi0 = σ and

linear order σ ≺ a ≺ b · · · .

• two letters x and y form a cover pair if x y and

∃z = x, y such that x ≺ z ≺ y.

• two letters xih and xih+1 , h ≥ 0 are a jump on x and y in

if x and y is not a cover pair, and either xih = x, xih+1 = y or xih = y, xih+1 = x (sums mod s + 1). An s-word is jump-free if it contains no jump for any {x, y} = ǫ. Theorem If

Jump-freeness

An s-word induces a finite chain σ ≺ a ≺ b ≺ c · · · . Given

i1 ǫxλ2 i2 ǫxλ3 i3 · · · ǫxλs is ] s-word with xi0 = σ and

linear order σ ≺ a ≺ b · · · .

• two letters x and y form a cover pair if x y and

∃z = x, y such that x ≺ z ≺ y.

• two letters xih and xih+1 , h ≥ 0 are a jump on x and y in

if x and y is not a cover pair, and either xih = x, xih+1 = y or xih = y, xih+1 = x (sums mod s + 1). An s-word is jump-free if it contains no jump for any {x, y} = ǫ. Theorem If

Theorem An s-word on an alphabet Σ is a sunword if and only if it is jump-free and satisfies the parity conditions.

ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ

ǫ ǫ

ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ σ σ a a b b b b c c c c a a ǫ σa2ǫb2ǫc2 σaǫbǫc2ǫbǫa

The characterization

Theorem Let G be a multisun. The following statements are equivalent (1) G is a sunoid (2) The s-word of G is a sunword (3) G is HOH-free Sketch of proof Since HOH-free multisuns satisfies the N-conditions hereditarily, we have that (3)⇒(1). By definition, (1)⇒(2). It remains to show that (2)⇒(3). Parity conditions and jump-freeness are crucial to prove that the sunoid represented by a sunword is HOH-free.

The characterization

Theorem Let G be a multisun. The following statements are equivalent (1) G is a sunoid (2) The s-word of G is a sunword (3) G is HOH-free Sketch of proof Since HOH-free multisuns satisfies the N-conditions hereditarily, we have that (3)⇒(1). By definition, (1)⇒(2). It remains to show that (2)⇒(3). Parity conditions and jump-freeness are crucial to prove that the sunoid represented by a sunword is HOH-free.

Characterization of MNB diamond-free graphs

Given a cycle C, the edges of C that form a triangle with another vertex of C are called non-proper. Odd generalized sun is a graph G whose vertex set can be partitioned into two sets: ⋄ a (not necessarily induced) odd cycle C of G with a set of non-proper edges {ej}j∈J (J is allowed to be empty) ⋄ a stable set U = {uj}j∈J such that uj is adjacent only to the endpoints of a non-proper edge of C. Theorem (Bonomo, Chudnovsky, Duran 2007) Let G be a diamond-free graph. Then G is clique-perfect if and only if no induced subgraph of G is an

• dd generalized sun.

Characterization of MNB diamond-free graphs

Given a cycle C, the edges of C that form a triangle with another vertex of C are called non-proper. Odd generalized sun is a graph G whose vertex set can be partitioned into two sets: ⋄ a (not necessarily induced) odd cycle C of G with a set of non-proper edges {ej}j∈J (J is allowed to be empty) ⋄ a stable set U = {uj}j∈J such that uj is adjacent only to the endpoints of a non-proper edge of C. Theorem (Bonomo, Chudnovsky, Duran 2007) Let G be a diamond-free graph. Then G is clique-perfect if and only if no induced subgraph of G is an

• dd generalized sun.

They noticed explicitly that the above one is far from being a minimal characterization. Indeed, the minimal characterization is: Theorem (G., Apollonio 2014) Let G be a diamond-free graph. Then G is clique-perfect if and only if no induced subgraph of G is an

• dd hole or a sunoid.

In other words: Theorem (G., Apollonio 2014) Let G be a diamond-free perfect graph. Then G is minimally non-balanced if and only if G is a sunoid.

They noticed explicitly that the above one is far from being a minimal characterization. Indeed, the minimal characterization is: Theorem (G., Apollonio 2014) Let G be a diamond-free graph. Then G is clique-perfect if and only if no induced subgraph of G is an

• dd hole or a sunoid.

In other words: Theorem (G., Apollonio 2014) Let G be a diamond-free perfect graph. Then G is minimally non-balanced if and only if G is a sunoid.

Thank you for your attention!!!

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