arXiv subjects

Nicolas Trotignon

Publications and source records attributed to Nicolas Trotignon.

At least 73 records · Page 4Linked to original sources

arXiv2013

Linear balanceable and subcubic balanceable graphs

In [{Structural properties and decomposition of linear balanced matrices}, {\it Mathematical Programming}, 55:129--168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also known as linear balanced bipartite graphs), and for balanced bipartite graphs whose maximum degree is at most 3. We in fact obtain results for more general classes, namely linear balanceable and subcubic balanceable graphs. Additionally, we prove that cubic balanced graphs contain a pair of twins, a result that was conjectured by Morris, Spiga and Webb in [Balanced Cayley graphs and balanced planar graphs, {\it Discrete Mathematics}, 310:3228--3235, 2010].

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arXiv2013

On graphs with no induced subdivision of $K_4$

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 $\cal C$ 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 $\cal C$ is 3-colorable and entails a polynomial-time recognition algorithm for membership in $\cal C$. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless.

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arXiv2013

Detecting an induced net subdivision

A {\em net} is a graph consisting of a triangle $C$ and three more vertices, each of degree one and with its neighbour in $C$, and all adjacent to different vertices of $C$. We give a polynomial-time algorithm to test whether an input graph has an induced subgraph which is a subdivision of a net. Unlike many similar questions, this does not seem to be solvable by an application of the "three-in-a-tree" subroutine.

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arXiv2013

Edge-colouring and total-colouring chordless graphs

A graph $G$ is \emph{chordless} if no cycle in $G$ has a chord. In the present work we investigate the chromatic index and total chromatic number of chordless graphs. We describe a known decomposition result for chordless graphs and use it to establish that every chordless graph of maximum degree $Δ\geq 3$ has chromatic index $Δ$ and total chromatic number $Δ+ 1$. The proofs are algorithmic in the sense that we actually output an optimal colouring of a graph instance in polynomial time.

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arXiv2013

Graphs that do not contain a cycle with a node that has at least two neighbors on it

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes, as well as polynomial time algorithms for vertex-coloring and edge-coloring.

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arXiv2013

Combinatorial optimization with 2-joins

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. We also give a simple class of graphs decomposable by 2-joins into bipartite graphs and line graphs, and for which finding a maximum stable set is NP-hard. This shows that having holes all of the same parity gives essential properties for the use of 2-joins in computing stable sets.

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arXiv2013

Detecting 2-joins faster

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 $O(n^3m)$ time where $n$ is the number of vertices of the input graph and $m$ the number of its edges. To detect \emph{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 $O(n^4m)$. Here, we give an $O(n^2m)$-time algorithm for both of these problems. A consequence is a speed up of several known algorithms.

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arXiv2013

Finding an induced subdivision of a digraph

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 NP-completeness proofs.

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arXiv2013

Excluding induced subdivisions of the bull and related graphs

For any graph $H$, let ${\rm Forb}^*(H)$ be the class of graphs with no induced subdivision of $H$. It was conjectured in [A.D. Scott, Induced trees in graphs of large chromatic number, {\em Journal of Graph Theory}, 24:297--311, 1997] that, for every graph $H$, there is a function $f_H:\mathbb{N} \rightarrow \mathbb{R}$ such that for every graph $G \in {\rm Forb}^*(H)$, $χ(G) \leq f_H(ω(G))$. We prove this conjecture for several graphs $H$, namely the paw (a triangle with a pendant edge), the bull (a triangle with two vertex-disjoint pendant edges), and what we call a "necklace," that is, a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge.

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arXiv2013

The $k$-in-a-tree problem for graphs of girth at least~$k$

For all integers $k\geq 3$, we give an $O(n^4)$ time algorithm for the problem whose instance is a graph $G$ of girth at least $k$ together with $k$ vertices and whose question is "Does $G$ contains an induced subgraph containing the $k$ vertices and isomorphic to a tree?". This directly follows for $k=3$ from the three-in-a-tree algorithm of Chudnovsky and Seymour and for $k=4$ from a result of Derhy, Picouleau and Trotignon. Here we solve the problem for $k\geq 5$. Our algorithm relies on a structural description of graphs of girth at least $k$ that do not contain an induced tree covering $k$ given vertices ($k\geq 5$).

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arXiv2013

On Roussel-Rubio-type lemmas and their consequences

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.

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arXiv2013

A structure theorem for graphs with no cycle with a unique chord and its consequences

We give a structural description of the class $\cal C$ 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 $\cal C$ is either in some simple basic class or has a decomposition. Basic classes are chordless cycles, cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraphs 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 ${\cal C}$, i.e. every graph in ${\cal C}$ 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 ${\cal C}$. This has several consequences: an ${\cal O}(nm)$-time algorithm to decide whether a graph is in $\cal C$, an ${\cal O}(n+m)$-time algorithm that finds a maximum clique of any graph in $\cal C$ and an ${\cal O}(nm)$-time coloring algorithm for graphs in $\cal C$. We prove that every graph in $\cal C$ is either 3-colorable or has a coloring with $ω$ colors where $ω$ is the size of a largest clique. The problem of finding a maximum stable set for a graph in $\cal C$ is known to be NP-hard.

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arXiv2013

The four-in-a-tree problem in triangle-free graphs

The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time $O(n^4)$ whether three given vertices of a graph belong to an induced tree. Here, we study four-in-a-tree for triangle-free graphs. We give a structural answer to the following question: what does a triangle-free graph look like if no induced tree covers four given vertices? Our main result says that any such graph must have the "same structure", in a sense to be defined precisely, as a square or a cube. We provide an $O(nm)$-time algorithm that given a triangle-free graph $G$ together with four vertices outputs either an induced tree that contains them or a partition of $V(G)$ certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree $T$ covering the four vertices such that at most one vertex of $T$ has degree at least 3 is NP-complete.

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arXiv2013

Detecting induced subgraphs

An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible} edges and \emph{real} edges. A \emph{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. Our NP-completeness proofs for $Π_B$'s rely on the NP-completeness proof of the following problem. Let $\cal S$ be a set of graphs and $d$ be an integer. Let $Γ_{\cal S}^d$ be the problem whose instance is $(G, x, y)$ where $G$ is a graph whose maximum degree is at most d, with no induced subgraph in $\cal S$ and $x, y \in V(G)$ are two non-adjacent vertices of degree 2. The question is "Does $G$ contain an induced cycle passing through $x, y$?". Among several results, we prove that $Γ^3_{\emptyset}$ is NP-complete. We give a simple criterion on a connected graph $H$ to decide whether $Γ^{+\infty}_{\{H\}}$ is polynomial or NP-complete. The polynomial cases rely on the algorithm three-in-a-tree, due to Chudnovsky and Seymour.

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arXiv2013

Decomposing Berge graphs and detecting balanced skew partitions

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class, or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One of them restricts the allowed decompositions to 2-joins and balanced skew partitions. We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an $O(n^9)$-time algorithm for the same problem restricted to Berge graphs. Our algorithm is not constructive: it only certifies whether a graph has a balanced skew partition or not. It relies on a new decomposition theorem for Berge graphs that is more precise than the previously known theorems. Our theorem also implies that every Berge graph can be decomposed in a first step by using only balanced skew partitions, and in a second step by using only 2-joins. Our proof of this new theorem uses at an essential step one of the theorems of Chudnovsky.

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arXiv2013

Algorithms for square-$3PC(\cdot, \cdot)$-free Berge graphs

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 claw-free 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.

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arXiv2013

Odd pairs of cliques

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 $K_1 \cup K_2$ 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 conjecture that a minimal imperfect graph has no odd pair of maximal cliques. We prove these conjectures in some special cases. 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.

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arXiv2013

A class of perfectly contractile graphs

We consider the class ${\cal A}$ 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 $G\in{\cal A}$ 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 [see the chapter "Even pairs" in the book {\it Perfect Graphs}, J.L. Ram\'ırez-Alfons\'ın and B.A. Reed, eds., Wiley Interscience, 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 ${\cal A}$. This entails a polynomial-time algorithm, based on successively contracting even pairs, to color optimally every graph in ${\cal A}$. This generalizes several results concerning some classical families of perfect graphs.

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