Rémy Belmonte

Graph classes with structured neighborhoods and algorithmic applications

Given a graph in any of the following graph classes: trapezoid graphs, circular permutation graphs, convex graphs, Dilworth k graphs, k-polygon graphs, circular arc graphs and complements of k-degenerate graphs, we show how to compute decompositions with the number of d-neighborhoods bounded by a polynomial of the input size. Combined with results of Bui-Xuan, Telle and Vatshelle (2013) [1] this leads to polynomial time algorithms for a large class of locally checkable vertex subset and vertex partitioning problems on all of these graph classes. The boolean-width of a graph is related to the number of 1-neighborhoods and our results imply that any of these graph classes have boolean-width O(logn).

Finding contractions and induced minors in chordal graphs via disjoint paths

The k-Disjoint Paths problem, which takes as input a graph G and k pairs of specified vertices (si,ti), asks whether G contains k mutually vertex-disjoint paths Pi such that Pi connects si and ti, for i=1,…,k. We study a natural variant of this problem, where the vertices of Pi must belong to a specified vertex subset Ui for i=1,…,k. In contrast to the original problem, which is polynomial-time solvable for any fixed integer k, we show that this variant is NP-complete even for k=2. On the positive side, we prove that the problem becomes polynomial-time solvable for any fixed integer k if the input graph is chordal. We use this result to show that, for any fixed graph H, the problems H-Contractibility and H-Induced Minor can be solved in polynomial time on chordal graphs. These problems are to decide whether an input graph G contains H as a contraction or as an induced minor, respectively.

Detecting Fixed Patterns in Chordal Graphs in Polynomial Time

The Contractibility problem takes as input two graphs G and H, and the task is to decide whether H can be obtained from G by a sequence of edge contractions. The Induced Minor and Induced Topological Minor problems are similar, but the first allows both edge contractions and vertex deletions, whereas the latter allows only vertex deletions and vertex dissolutions. All three problems are NP-complete, even for certain fixed graphs H. We show that these problems can be solved in polynomial time for every fixed H when the input graph G is chordal. Our results can be considered tight, since these problems are known to be W[1]-hard on chordal graphs when parameterized by the size of H. To solve Contractibility and Induced Minor, we define and use a generalization of the classic Disjoint Paths problem, where we require the vertices of each of the k paths to be chosen from a specified set. We prove that this variant is NP-complete even when k=2, but that it is polynomial-time solvable on chordal graphs for every fixed k. Our algorithm for Induced Topological Minor is based on another generalization of Disjoint Paths called Induced Disjoint Paths, where the vertices from different paths may no longer be adjacent. We show that this problem, which is known to be NP-complete when k=2, can be solved in polynomial time on chordal graphs even when k is part of the input. Our results fit into the general framework of graph containment problems, where the aim is to decide whether a graph can be modified into another graph by a sequence of specified graph operations. Allowing combinations of the four well-known operations edge deletion, edge contraction, vertex deletion, and vertex dissolution results in the following ten containment relations: (induced) minor, (induced) topological minor, (induced) subgraph, (induced) spanning subgraph, dissolution, and contraction. Our results, combined with existing results, settle the complexity of each of the ten corresponding containment problems on chordal graphs.

Edge contractions in subclasses of chordal graphs

Modifying a given graph to obtain another graph is a well-studied problem with applications in many fields. Given two input graphs G and H, the Contractibility problem is to decide whether H can be obtained from G by a sequence of edge contractions. This problem is known to be NP-complete already when both input graphs are trees of bounded diameter. We prove that Contractibility can be solved in polynomial time when G is a trivially perfect graph and H is a threshold graph, thereby giving the first classes of graphs of unbounded treewidth and unbounded degree on which the problem can be solved in polynomial time. We show that this polynomial-time result is in a sense tight, by proving that Contractibility is NP-complete when G and H are both trivially perfect graphs, and when G is a split graph and H is a threshold graph. If the graph H is fixed and only G is given as input, then the problem is called H-Contractibility. This problem is known to be NP-complete on general graphs already when H is a path on four vertices. We show that, for any fixed graph H, the H-Contractibility problem can be solved in polynomial time if the input graph G is a split graph.

The price of connectivity for feedback vertex set

Let fvs ( G ) and cfvs ( G ) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of aźgraph G , respectively. The price of connectivity for feedback vertex set (poc-fvs) for a class of graphs G is defined as the maximum ratio cfvs ( G ) / fvs ( G ) over all connected graphs G ź G . We study the poc-fvs for graph classes defined by a finite family H of forbidden induced subgraphs. We characterize exactly those finite families H for which the poc-fvs for H -free graphs is upper bounded by a constant. Additionally, for the case where ź H ź = 1 , we determine exactly those graphs H for which there exists a constant c H such that cfvs ( G ) ź fvs ( G ) + c H for every connected H -free graph G , as well as exactly those graphs H for which we can take c H = 0 .

Graph classes and Ramsey numbers

Abstract For a graph class G and any two positive integers i and j , the Ramsey number R G ( i , j ) is the smallest positive integer such that every graph in G on at least R G ( i , j ) vertices has a clique of size i or an independent set of size j . For the class of all graphs, Ramsey numbers are notoriously hard to determine, and they are known only for very small values of i and j . Even if we restrict G to be the class of claw-free graphs, it is highly unlikely that a formula for determining R G ( i , j ) for all values of i and j will ever be found, as there are infinitely many nontrivial Ramsey numbers for claw-free graphs that are as difficult to determine as for arbitrary graphs. Motivated by this difficulty, we establish here exact formulas for all Ramsey numbers for three important subclasses of claw-free graphs: line graphs, long circular interval graphs, and fuzzy circular interval graphs. On the way to obtaining these results, we also establish all Ramsey numbers for the class of perfect graphs. Such positive results for graph classes are rare: a formula for determining R G ( i , j ) for all values of i and j , when G is the class of planar graphs, was obtained by Steinberg and Tovey (1993), and this seems to be the only previously known result of this kind. We complement our aforementioned results by giving exact formulas for determining all Ramsey numbers for several graph classes related to perfect graphs.

Forbidden Induced Subgraphs and the Price of Connectivity for Feedback Vertex Set

Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. For a graph class \({\cal G}\), the price of connectivity for feedback vertex set (poc-fvs) for \({\cal G}\) is defined as the maximum ratio cfvs(G)/fvs(G) over all connected graphs G in \({\cal G}\). It is known that the poc-fvs for general graphs is unbounded. We study the poc-fvs for graph classes defined by a finite family \({\cal H}\) of forbidden induced subgraphs. We characterize exactly those finite families \({\cal H}\) for which the poc-fvs for \({\cal H}\)-free graphs is bounded by a constant. Prior to our work, such a result was only known for the case where \(|{\cal H}|=1\).

Parameterized Complexity of Two Edge Contraction Problems with Degree Constraints

Motivated by recent results of Mathieson and Szeider (J. Comput. Syst. Sci. 78(1): 179–191, 2012), we study two graph modification problems where the goal is to obtain a graph whose vertices satisfy certain degree constraints. The Regular Contraction problem takes as input a graph G and two integers d and k, and the task is to decide whether G can be modified into a d-regular graph using at most k edge contractions. The Bounded Degree Contraction problem is defined similarly, but here the objective is to modify G into a graph with maximum degree at most d. We observe that both problems are fixed-parameter tractable when parameterized jointly by k and d. We show that when only k is chosen as the parameter, Regular Contraction becomes W[1]-hard, while Bounded Degree Contraction becomes W[2]-hard even when restricted to split graphs. We also prove both problems to be NP-complete for any fixed d ≥ 2. On the positive side, we show that the problem of deciding whether a graph can be modified into a cycle using at most k edge contractions, which is equivalent to Regular Contraction when d = 2, admits an O(k) vertex kernel. This complements recent results stating that the same holds when the target is a path, but that the problem admits no polynomial kernel when the target is a tree, unless NP ⊆ coNP/poly (Heggernes et al., IPEC 2011).

The price of connectivity for feedback vertex set.

Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. In general graphs, the ratio cfvs(G)/fvs(G) can be arbitrarily large. We study the interdependence between fvs(G) and cfvs(G) in graph classes defined by excluding one induced subgraph H. We show that the ratio cfvs(G)/fvs(G) is bounded by a constant for every connected H-free graph G if and only if H is a linear forest. We also determine exactly those graphs H for which there exists a constant c H such that cfvs(G) ≤ fvs(G) + c H for every connected H-free graph G, as well as exactly those graphs H for which we can take c H = 0.

Metric Dimension of Bounded Width Graphs

The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y. A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph. The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes. In recent work, Foucaud et al. (2015) showed that this problem is NP-complete even on interval graphs. They complemented this result by also showing that it is fixed-parameter tractable (FPT) parameterized by the metric dimension of the graph. In this work, we show that this FPT result can in fact be extended to all graphs of bounded tree-length. This includes well-known classes like chordal graphs, AT-free graphs and permutation graphs. We also show that this problem is FPT parameterized by the modular-width of the input graph.