Nicolas Trotignon | École Normale Supérieure de Lyon (original) (raw)

Papers by Nicolas Trotignon

... Graphes parfaits : structure et algorithmes. ... de coloration des graphes Artémis de complex... more ... Graphes parfaits : structure et algorithmes. ... de coloration des graphes Artémis de complexité O(mn^2). Nous donnons un algorithme de complexité O(n^9 ... D'autres algorithmes de reconnaissance sont donnés, tous fondés sur des routines de détection de sous-graphes dans ...

Journal of Combinatorial Theory

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs c... more A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset.

A graph is Berge if it has no induced odd cycle on at least 5 vertices and no complement of induc... more A graph is Berge if it has no induced odd cycle on at least 5 vertices and no complement of induced odd cycle on at least 5 vertices. A graph is perfect if the chromatic number equals the maximum clique number for every induced subgraph. Chudnovsky, Robertson, Seymour and Thomas proved that every Berge graph either falls into some classical family of perfect graphs, or has a structural fault that cannot occur in a minimal imperfect graph. A corollary of this is the strong perfect graph theorem conjectured by Berge: every Berge graph is perfect. An even pair of vertices in a graph is a pair of vertices such that every induced path between them has even length. Meyniel proved that a minimal imperfect graph cannot contain an even pair. So even pairs may be considered as a structural fault. Chudnovsky et al. do not use them, and it is known that some classes of Berge graph have no even pairs. The aim of this work is to investigate an “even-pair-like” notion that could be a structural fault present in every Berge graph. An odd pair of cliques is a pair of cliques {K 1, K 2} such that every induced path from K 1 to K 2 with no interior vertex in K 1 ∪ K 2 has odd length. We conjecture that for every Berge graph G on at least two vertices, either one of G, \( \bar G \) has an even pair, or one of G, \( \bar G \) has an odd pair of cliques. We note that this conjecture is true for basic perfect graphs. By the strong perfect graph theorem, we know that a minimal imperfect graph has no odd pair of maximal cliques. In some special cases we prove this fact independently of the strong perfect graph theorem. We show that adding all edges between any 2 vertices of the cliques of an odd pair of cliques is an operation that preserves perfectness.

Journal of Combinatorial Theory, 2006

We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph ... more We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them). We prove that every graph G ∈ A different from a clique has an "even pair" (two vertices that are not joined by a chordless path of odd length), as conjectured by Everett and Reed ["Even pairs", in: J.L. Ramírez-Alfonsín, B.A. Reed (eds.), Perfect Graphs, Wiley Interscience, New York, 2001]. Our proof is a polynomial-time algorithm that produces an even pair with the additional property that the contraction of this pair yields a graph in A. This entails a polynomial-time algorithm, based on successively contracting even pairs, to color optimally every graph in A. This generalizes several results concerning some classical families of perfect graphs.

Electronic Notes in Discrete Mathematics, 2005

We consider the class of graphs containing no odd hole, no odd antihole, and no configuration con... more We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole, and at least two of the paths are of length 2. This class generalizes clawfree Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of complexity O(n 7 ) to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class.

Let F be the tree we obtain from the claw (K1,3) by subdividing two of its three branches once, a... more Let F be the tree we obtain from the claw (K1,3) by subdividing two of its three branches once, and twice, respectively. We show that every graph with no triangle and no induced F is 8-colourable.

We prove a decomposition theorem for graphs that do not contain a subdivision of K 4 as an induce... more We prove a decomposition theorem for graphs that do not contain a subdivision of K 4 as an induced subgraph where K 4 is the complete graph on four vertices. We obtain also a structure theorem for the class C of graphs that contain neither a subdivision of K 4 nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in C is 3-colorable and entails a polynomial-time recognition algorithm for membership in C. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless. q } be the two sides of the bipartition of H . If v is adjacent to at most one vertex in A and at most one in B, then the lemma holds. Suppose now, up to symmetry, that v is adjacent to at least two vertices in A, say a 1 , a 2 . Then v is either adjacent to every vertex in B or to no vertex in B, for otherwise, up to symmetry, v is adjacent to b 1 and not to b 2 , and {a 1 ,

Computing Research Repository, 2005

We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph ... more We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n 2 m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper.

Discrete Applied Mathematics, 2009

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation o... more An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary length (at least one). Given an s-graph B, we study the decision problem Π B whose instance is a graph G and question is ''Does G contain a realisation of B as an induced subgraph?''. For several B's, the complexity of Π B is known and here we give the complexity for several more.

Discrete Mathematics, 2011

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theo... more Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward's decomposition theorem for weakly chordal graphs, relying on a Roussel-Rubio-type lemma. We recall how Roussel-Rubio-type lemmas yield very short proofs of the existence of even pairs in weakly chordal graphs and Meyniel graphs.

Electronic Notes in Discrete Mathematics, 2011

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digr... more We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) G, does it contain an induced subdivision of a prescribed digraph D? The complexity of this problem depends on D and on whether G must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NPcompleteness proofs.

Journal of Graph Theory, 2010

We give a structural description of the class C of graphs that do not contain a cycle with a uniq... more We give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is a either in some simple basic class or has a decomposition. Basic classes are cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraph of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations; and all graphs built this way are in C.

Theoretical Computer Science, 2009

We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph ... more We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n 2 m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper.

Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs that do not contain a subdivisio... more Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. Documents de

2-joins are edge cutsets that naturally appear in the decomposition of several classes of graphs ... more 2-joins are edge cutsets that naturally appear in the decomposition of several classes of graphs closed under taking induced subgraphs, such as balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free graphs. Their detection is needed in several algorithms, and is the slowest step for some of them. The classical method to detect a 2-join takes O(n 3 m) time where n is the number of vertices of the input graph and m the number of its edges. To detect non-path 2-joins (special kinds of 2-joins that are needed in all of the known algorithms that use 2-joins), the fastest known method takes time O(n 4 m). Here, we give an O(n 2 m)-time algorithm for both of these problems. A consequence is a speed up of several known algorithms.

Siam Journal on Discrete Mathematics, 2005

We consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and... more We consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and no \prism" (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class A0 of graphs that contain no odd hole, no antihole of length at least 5, and no odd prism (prism whose three

Electronic Notes in Discrete Mathematics, 2007

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation o... more An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary length (at least one). Given an s-graph B, we study the decision problem Π B whose instance is a graph G and question is "Does G contain a realisation of B as an induced subgraph ?". For several B's, the complexity of Π B is known and here we give the complexity for several more.

Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs that do not contain a subdivisio... more Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. Documents de

European Journal of Combinatorics, 2004

We say that a 0-1 matrix N of size a × b can be found in a collection of sets H if we can find se... more We say that a 0-1 matrix N of size a × b can be found in a collection of sets H if we can find sets H 1 , H 2 , . . . , H a in H and elements e 1 , e 2 , . . . , e b in ∪ H∈H H such that N is the incidence matrix of the sets H 1 , H 2 , . . . , H a over the elements e 1 , e 2 , . . . , e b . We prove the following Ramsey-type result: for every n ∈ N, there exists a number S(n) such that in any collection of at least S(n) sets, one can find either the incidence matrix of a collection of n singletons, or its complementary matrix, or the incidence matrix of a collection of n sets completely ordered by inclusion. We give several results of the same extremal set theoretical flavour. For some of these, we give the exact value of the number of sets required.

... Graphes parfaits : structure et algorithmes. ... de coloration des graphes Artémis de complex... more ... Graphes parfaits : structure et algorithmes. ... de coloration des graphes Artémis de complexité O(mn^2). Nous donnons un algorithme de complexité O(n^9 ... D'autres algorithmes de reconnaissance sont donnés, tous fondés sur des routines de détection de sous-graphes dans ...

Journal of Combinatorial Theory

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs c... more A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset.

A graph is Berge if it has no induced odd cycle on at least 5 vertices and no complement of induc... more A graph is Berge if it has no induced odd cycle on at least 5 vertices and no complement of induced odd cycle on at least 5 vertices. A graph is perfect if the chromatic number equals the maximum clique number for every induced subgraph. Chudnovsky, Robertson, Seymour and Thomas proved that every Berge graph either falls into some classical family of perfect graphs, or has a structural fault that cannot occur in a minimal imperfect graph. A corollary of this is the strong perfect graph theorem conjectured by Berge: every Berge graph is perfect. An even pair of vertices in a graph is a pair of vertices such that every induced path between them has even length. Meyniel proved that a minimal imperfect graph cannot contain an even pair. So even pairs may be considered as a structural fault. Chudnovsky et al. do not use them, and it is known that some classes of Berge graph have no even pairs. The aim of this work is to investigate an “even-pair-like” notion that could be a structural fault present in every Berge graph. An odd pair of cliques is a pair of cliques {K 1, K 2} such that every induced path from K 1 to K 2 with no interior vertex in K 1 ∪ K 2 has odd length. We conjecture that for every Berge graph G on at least two vertices, either one of G, \( \bar G \) has an even pair, or one of G, \( \bar G \) has an odd pair of cliques. We note that this conjecture is true for basic perfect graphs. By the strong perfect graph theorem, we know that a minimal imperfect graph has no odd pair of maximal cliques. In some special cases we prove this fact independently of the strong perfect graph theorem. We show that adding all edges between any 2 vertices of the cliques of an odd pair of cliques is an operation that preserves perfectness.

Journal of Combinatorial Theory, 2006

We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph ... more We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them). We prove that every graph G ∈ A different from a clique has an "even pair" (two vertices that are not joined by a chordless path of odd length), as conjectured by Everett and Reed ["Even pairs", in: J.L. Ramírez-Alfonsín, B.A. Reed (eds.), Perfect Graphs, Wiley Interscience, New York, 2001]. Our proof is a polynomial-time algorithm that produces an even pair with the additional property that the contraction of this pair yields a graph in A. This entails a polynomial-time algorithm, based on successively contracting even pairs, to color optimally every graph in A. This generalizes several results concerning some classical families of perfect graphs.

Electronic Notes in Discrete Mathematics, 2005

We consider the class of graphs containing no odd hole, no odd antihole, and no configuration con... more We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole, and at least two of the paths are of length 2. This class generalizes clawfree Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of complexity O(n 7 ) to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class.

Let F be the tree we obtain from the claw (K1,3) by subdividing two of its three branches once, a... more Let F be the tree we obtain from the claw (K1,3) by subdividing two of its three branches once, and twice, respectively. We show that every graph with no triangle and no induced F is 8-colourable.

We prove a decomposition theorem for graphs that do not contain a subdivision of K 4 as an induce... more We prove a decomposition theorem for graphs that do not contain a subdivision of K 4 as an induced subgraph where K 4 is the complete graph on four vertices. We obtain also a structure theorem for the class C of graphs that contain neither a subdivision of K 4 nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in C is 3-colorable and entails a polynomial-time recognition algorithm for membership in C. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless. q } be the two sides of the bipartition of H . If v is adjacent to at most one vertex in A and at most one in B, then the lemma holds. Suppose now, up to symmetry, that v is adjacent to at least two vertices in A, say a 1 , a 2 . Then v is either adjacent to every vertex in B or to no vertex in B, for otherwise, up to symmetry, v is adjacent to b 1 and not to b 2 , and {a 1 ,

Computing Research Repository, 2005

We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph ... more We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n 2 m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper.

Discrete Applied Mathematics, 2009

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation o... more An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary length (at least one). Given an s-graph B, we study the decision problem Π B whose instance is a graph G and question is ''Does G contain a realisation of B as an induced subgraph?''. For several B's, the complexity of Π B is known and here we give the complexity for several more.

Discrete Mathematics, 2011

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theo... more Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward's decomposition theorem for weakly chordal graphs, relying on a Roussel-Rubio-type lemma. We recall how Roussel-Rubio-type lemmas yield very short proofs of the existence of even pairs in weakly chordal graphs and Meyniel graphs.

Electronic Notes in Discrete Mathematics, 2011

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digr... more We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) G, does it contain an induced subdivision of a prescribed digraph D? The complexity of this problem depends on D and on whether G must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NPcompleteness proofs.

Journal of Graph Theory, 2010

We give a structural description of the class C of graphs that do not contain a cycle with a uniq... more We give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is a either in some simple basic class or has a decomposition. Basic classes are cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraph of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations; and all graphs built this way are in C.

Theoretical Computer Science, 2009

We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph ... more We consider the class A of graphs that contain no odd hole, no antihole, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n 2 m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper.

Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs that do not contain a subdivisio... more Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. Documents de

2-joins are edge cutsets that naturally appear in the decomposition of several classes of graphs ... more 2-joins are edge cutsets that naturally appear in the decomposition of several classes of graphs closed under taking induced subgraphs, such as balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free graphs. Their detection is needed in several algorithms, and is the slowest step for some of them. The classical method to detect a 2-join takes O(n 3 m) time where n is the number of vertices of the input graph and m the number of its edges. To detect non-path 2-joins (special kinds of 2-joins that are needed in all of the known algorithms that use 2-joins), the fastest known method takes time O(n 4 m). Here, we give an O(n 2 m)-time algorithm for both of these problems. A consequence is a speed up of several known algorithms.

Siam Journal on Discrete Mathematics, 2005

We consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and... more We consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and no \prism" (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class A0 of graphs that contain no odd hole, no antihole of length at least 5, and no odd prism (prism whose three

Electronic Notes in Discrete Mathematics, 2007

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation o... more An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary length (at least one). Given an s-graph B, we study the decision problem Π B whose instance is a graph G and question is "Does G contain a realisation of B as an induced subgraph ?". For several B's, the complexity of Π B is known and here we give the complexity for several more.

Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs that do not contain a subdivisio... more Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. Documents de

European Journal of Combinatorics, 2004

We say that a 0-1 matrix N of size a × b can be found in a collection of sets H if we can find se... more We say that a 0-1 matrix N of size a × b can be found in a collection of sets H if we can find sets H 1 , H 2 , . . . , H a in H and elements e 1 , e 2 , . . . , e b in ∪ H∈H H such that N is the incidence matrix of the sets H 1 , H 2 , . . . , H a over the elements e 1 , e 2 , . . . , e b . We prove the following Ramsey-type result: for every n ∈ N, there exists a number S(n) such that in any collection of at least S(n) sets, one can find either the incidence matrix of a collection of n singletons, or its complementary matrix, or the incidence matrix of a collection of n sets completely ordered by inclusion. We give several results of the same extremal set theoretical flavour. For some of these, we give the exact value of the number of sets required.