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¨ onig 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¨ onig property if and only if ν ( A ) = τ ( A ) . 1 1 0 0 0 · · · 0 1 1 0 0 odd cycle matrix has · · · ... ν ( C 2 k + 1 ) = k < τ ( C 2 k + 1 ) = k + 1 C 2 k + 1 = 0 0 1 0 0 . . . ... . . . . . . 1 1 1 0 0 0 1 · · ·
K¨ onig 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¨ onig property if and only if ν ( A ) = τ ( A ) . 1 1 0 0 0 · · · 0 1 1 0 0 odd cycle matrix has · · · ... ν ( C 2 k + 1 ) = k < τ ( C 2 k + 1 ) = k + 1 C 2 k + 1 = 0 0 1 0 0 . . . ... . . . . . . 1 1 1 0 0 0 1 · · ·
K¨ onig 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¨ onig property if and only if ν ( A ) = τ ( A ) . 1 1 0 0 0 · · · 0 1 1 0 0 odd cycle matrix has · · · ... ν ( C 2 k + 1 ) = k < τ ( C 2 k + 1 ) = k + 1 C 2 k + 1 = 0 0 1 0 0 . . . ... . . . . . . 1 1 1 0 0 0 1 · · ·
K¨ onig 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¨ onig property if and only if ν ( A ) = τ ( A ) . 1 1 0 0 0 · · · 0 1 1 0 0 odd cycle matrix has · · · ... ν ( C 2 k + 1 ) = k < τ ( C 2 k + 1 ) = k + 1 C 2 k + 1 = 0 0 1 0 0 . . . ... . . . . . . 1 1 1 0 0 0 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 �⊇ C 2 k + 1 , k ≥ 1 , as a submatrix G graph, A G 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 A G 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 �⊇ C 2 k + 1 , k ≥ 1 , as a submatrix G graph, A G 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 A G 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 �⊇ C 2 k + 1 , k ≥ 1 , as a submatrix G graph, A G 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 A G 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 �⊇ C 2 k + 1 , k ≥ 1 , as a submatrix G graph, A G 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 A G 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 odd 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 odd 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 odd 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 τ c ( G ) = min # of vertices that meets all the maximal cliques of G . α c ( G ) = max # of pairwise vertex-disjoint cliques of G . A graph G is clique-perfect if G ′ of G . τ c ( G ) = α c ( G ) for each induced subgraph 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 τ c ( G ) = min # of vertices that meets all the maximal cliques of G . α c ( G ) = max # of pairwise vertex-disjoint cliques of G . A graph G is clique-perfect if G ′ of G . τ c ( G ) = α c ( G ) for each induced subgraph 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 τ c ( G ) = min # of vertices that meets all the maximal cliques of G . α c ( G ) = max # of pairwise vertex-disjoint cliques of G . A graph G is clique-perfect if G ′ of G . τ c ( G ) = α c ( G ) for each induced subgraph 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 τ c ( G ) = min # of vertices that meets all the maximal cliques of G . α c ( G ) = max # of pairwise vertex-disjoint cliques of G . A graph G is clique-perfect if G ′ of G . τ c ( G ) = α c ( G ) for each induced subgraph 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 τ c ( G ) = min # of vertices that meets all the maximal cliques of G . α c ( G ) = max # of pairwise vertex-disjoint cliques of G . A graph G is clique-perfect if G ′ of G . τ c ( G ) = α c ( G ) for each induced subgraph 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 τ c ( G ) = min # of vertices that meets all the maximal cliques of G . α c ( G ) = max # of pairwise vertex-disjoint cliques of G . A graph G is clique-perfect if G ′ of G . τ c ( G ) = α c ( G ) for each induced subgraph 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.
Recommend
More recommend
