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Frédéric Maffray

Publications and source records attributed to Frédéric Maffray.

33 records · Page 2Linked to original sources
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.

cs.DM↗
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.

cs.DM↗
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.

math.CO↗
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.

math.CO↗
arXiv2013

Algorithms for perfectly contractile graphs

We consider the class ${\cal A}$ 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 ${\cal A}'$ of graphs that contain no odd hole, no antihole of length at least 5 and no odd prism (prism whose three paths are odd). These two classes were introduced by Everett and Reed and are relevant to the study of perfect graphs. We give polynomial-time recognition algorithms for these two classes. We proved previously that every graph $G\in{\cal A}$ is "perfectly contractile", 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]. The analogous conjecture concerning graphs in ${\cal A}'$ is still open.

cs.DM↗
arXiv2013

Ramsey-type results on singletons, co-singletons and monotone sequences in large collections of sets

We say that a 0-1 matrix $N$ of size $a\times b$ can be found in a collection of sets $\mathcal{H}$ if we can find sets $H_{1}, H_{2}, \dots, H_{a}$ in $\mathcal{H}$ and elements $e_1, e_2, \dots, e_b$ in $\cup_{H \in \mathcal{H}} H$ such that $N$ is the incidence matrix of the sets $H_{1}, H_{2}, \dots, H_{a}$ over the elements $e_1, e_2, \dots, e_b$. We prove the following Ramsey-type result: for every $n\in \N$, 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.

math.CO↗
arXiv2012

Fire Containment in Planar Graphs

In a graph $G$, a fire starts at some vertex. At every time step, firefighters can protect up to $k$ vertices, and then the fire spreads to all unprotected neighbours. The $k$-surviving rate $ρ_k(G)$ of $G$ is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs $\cG$ we are interested in the minimum value $k$ such that $ρ_k(G)\geε$ for some constant $ε>0$ and all $G\in\cG$ i.e., such that linearly many vertices are expected to be saved in every graph from $\cG$). In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.

math.CO↗
arXiv2010

A characterization of b-perfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph $G$ is the largest integer $k$ such that $G$ admits a b-coloring with $k$ colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph of $G$. We prove that a graph is b-perfect if and only if it does not contain as an induced subgraph a member of a certain list of twenty-two graphs. This entails the existence of a polynomial-time recognition algorithm and of a polynomial-time algorithm for coloring exactly the vertices of every b-perfect graph.

cs.DM↗
arXiv2008

Characterizing path graphs by forbidden induced subgraphs

A graph is a path graph if it is the intersection graph of a family of subpaths of a tree. In 1970, Renz asked for a characterizaton of path graphs by forbidden induced subgraphs. Here we answer this question by listing all graphs that are not path graphs and are minimal with this property.

cs.DM↗
arXiv2007

On b-perfect chordal graphs

The b-chromatic number of a graph G is the largest integer k such that G has a coloring of the vertices in k color classes such that every color class contains a vertex that has a neighbour in all other color classes. We characterize the class of chordal graphs for which the b-chromatic number is equal to the chromatic number for every induced subgraph.

cs.DM↗
arXiv2007

Coloring vertices of a graph or finding a Meyniel obstruction

A Meyniel obstruction is an odd cycle with at least five vertices and at most one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds either a clique and coloring of the same size or a Meyniel obstruction. We also give a O(n^3) algorithm that, for any graph, finds either aneasily recognizable strong stable set or a Meyniel obstruction.

cs.DM↗
arXiv2006

Erratum : MCColor is not optimal on Meyniel graphs

A Meyniel graph is a graph in which every odd cycle of length at least five has two chords. In the manuscript "Coloring Meyniel graphs in linear time" we claimed that our algorithm MCColor produces an optimal coloring for every Meyniel graph. But later we found a mistake in the proof and a couterexample to the optimality, which we present here. MCColor can still be used to find a stable set that intersects all maximal cliques of a Meyniel graph in linear time. Consequently it can be used to find an optimal coloring in time O(nm), and the same holds for Algorithm MCS+Color. This is explained in the manuscript "A linear algorithm to find a strong stable set in a Meyniel graph" but this is equivalent to Hertz's algorithm. The current best algorithm for coloring Meyniel graphs is the O(n^2) algorithm LexColor due to Roussel and Rusu. The question of finding a linear-time algorithm to color Meyniel graphs is still open.

cs.DM↗
arXiv2005

Precoloring co-Meyniel graphs

The pre-coloring extension problem consists, given a graph $G$ and a subset of nodes to which some colors are already assigned, in finding a coloring of $G$ with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. We answer a question of Hujter and Tuza by showing that ``PrExt perfect'' graphs are exactly the co-Meyniel graphs, which also generalizes results of Hujter and Tuza and of Hertz. Moreover we show that, given a co-Meyniel graph, the corresponding contracted graph belongs to a restricted class of perfect graphs (``co-Artemis'' graphs, which are ``co-perfectly contractile'' graphs), whose perfectness is easier to establish than the strong perfect graph theorem. However, the polynomiality of our algorithm still depends on the ellipsoid method for coloring perfect graphs.

cs.DM↗
arXiv2005

Coloring Artemis graphs

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^2m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper.

cs.DM↗