Computing the diameter, and more generally, all eccentricities of an undirected graph is an important problem in algorithmic graph theory and the challenge is to identify graph classes for which their computation can be achieved in subquadratic time. Using a new recursive scheme based on the structural properties of median graphs, we provide a quasilinear-time algorithm to determine all eccentricities for this well-known family of graphs. The gist of our technique is to identify the balanced and unbalanced parts of the Θ-class decomposition of median graphs, which are then processed using different recursive schemes. The exact running time of our algorithm is in O (n log4 n ). This outcome not only answers a question asked by Bénéteau et al. (2020) but also greatly improves the recent combinatorial algorithm of Berge et al. (2022) for the same problem, running in time O (n1.6408 logO (1) n ).

In the context of fine-grained complexity, we investigate the notion of certificate enabling faster polynomialtime algorithms. We specifically target radius (minimum eccentricity), diameter (maximum eccentricity), and all-eccentricity computations for which quadratic-time lower bounds are known under plausible conjectures. In each case, we introduce a notion of certificate as a specific set of nodes from which appropriate bounds on all eccentricities can be derived in subquadratic time when this set has sublinear size. The existence of small certificates is a barrier against SETH-based lower bounds for these problems. We indeed prove that for graph classes with small certificates, there exist randomized subquadratic-time algorithms for computing the radius, the diameter, and all eccentricities respectively.

International audience

Abstract. Modular decomposition focuses on repeatedly identifying a module [Formula: see text] (a collection of vertices that shares exactly the same neighborhood outside of [Formula: see text]) and collapsing it into a single vertex. This notion of exactitude of neighborhood is very strict, especially when dealing with real-world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an [Formula: see text]- module, a relaxation that maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal [Formula: see text]-modules can be computed in polynomial time, and we generalize series and parallel operation between graphs. This leads to [Formula: see text]-cographs which have interesting properties. We study how to generalize Gallai’s theorem corresponding to the case for [Formula: see text], but unfortunately we give evidence that computing such a decomposition tree can be difficult.

International audience

Abstract. Under the strong exponential-time hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time (in the size [Formula: see text] of the input), as shown by Roditty and Williams. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs, and planar graphs, to name a few. We propose to study unweighted graphs of constant distance Vapnik–Chervonenkis (VC)-dimension as a broad generalization of many such classes—where the distance VC-dimension of a graph [Formula: see text] is defined as the VC-dimension of its ball hypergraph whose hyperedges are the balls of all possible radii and centers in [Formula: see text]. In particular for any fixed [Formula: see text], the class of [Formula: see text]-minor free graphs has distance VC-dimension at most [Formula: see text]. Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most [Formula: see text], for any fixed [Formula: see text], either computes the diameter or concludes that it is larger than [Formula: see text] in time [Formula: see text], where [Formula: see text] only depends on [Formula: see text] and the [Formula: see text] notation suppresses polylogarithmic factors. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. Then as a byproduct of our approach, we get a truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs, and graphs of bounded expansion. Before our work, the only known such algorithm was resulting from an application of Courcelle’s theorem; see Grohe, Kreutzer, and Siebertz [ J. ACM, 64 (2017), pp. 1–32]. For any graph of constant distance VC-dimension, we further prove the existence of an exact distance oracle in truly subquadratic space, that answers distance queries in truly sublinear time (in the number [Formula: see text] of vertices). The latter generalizes prior results on proper minor-closed graph classes to a much larger graph class. Finally, we show how to remove the dependency on [Formula: see text] for any graph class that excludes a fixed graph [Formula: see text] as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time deterministic algorithm for computing all the eccentricities, and thus both the diameter and the radius. Our approach can be generalized to the [Formula: see text]-minor free graphs with bounded positive integer weights. We note that all our algorithms for the diameter problem can be adapted for computing the radius, and more generally all the eccentricities. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of [Formula: see text]-nets, region decomposition, and other partition techniques.

Median graphs form the class of graphs which is the most studied in metric graph theory. Recently, Bénéteau et al. [2019] designed a linear-time algorithm computing both the Θ-classes and the median set of median graphs. A natural question emerges: is there a linear-time algorithm computing the diameter for median graphs? We answer positively to this question for median graphs G with constant dimension d, i.e. the dimension of the largest induced hypercube of G. We propose a combinatorial algorithm computing the diameter of median graphs with running time O(2O(d log d)n). In particular, since the hypercube Q4 of dimension 4 is not planar, it shows also that the diameter of planar median graphs can be computed in O(n).

This corrigendum corrects errors found by Jérémie Dusart in the proof of correctness of the algorithms in [D. G. Corneil, B. Dalton, and M. Habib, SIAM J. Comput., 42 (2013), pp. 792--807]; there are no changes in the algorithms themselves.

This paper deals with the characterization and the recognition of graph classes. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately, this strategy rarely leads to a very efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques that use some ordering of the nodes, such as the one given by a traversal. We specifically study graphs that have an ordering avoiding some ordered structures. More precisely, we consider structures that we call patterns on three nodes, and we study the complexity of recognizing the classes associated with such patterns. In this domain, there are three key previous works. Independently Skrien [ J. Graph Theory, 6 (1982), pp. 309--316] and Damashke [Forbidden ordered subgraphs, in Topics in Combinatorics and Graph Theory, Physica-Verlag HD, 1990, pp. 219--229] noted that several graph classes, such as chordal, bipartite, interval, and comparability graphs, have a characterization in terms of forbidden patterns. On the algorithmic side, Hell, Mohar, and Rafiey [Ordering without forbidden patterns, in Algorithms--ESA 2014, Springer, 2014, pp. 554--565] proved that any class defined by a set of forbidden patterns on three nodes can be recognized in time $O(n^3)$ by using an algorithm based on an extension of 2-SAT. We improve on these two lines of works by systematically characterizing all the classes defined by sets of forbidden patterns (on three nodes) and proving that among the 22 different classes (up to complement) that we find, 20 can actually be recognized in linear time. Beyond these results, we consider that this type of characterization is very useful from an algorithmic perspective, leads to a rich structure of classes, and generates many algorithmic and structural open questions worth investigating.

Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes – where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V(H)| – 1. Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most d, for any fixed k, either computes the diameter or concludes that it is larger than k in time (k · mn1−εd), where εd ϵ (0; 1) only depends on d. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. Then as a byproduct of our approach, we get the first truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of bounded expansion. Finally, we show how to remove the dependency on k for any graph class that excludes a fixed graph H as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time randomized algorithm for computing their diameter. We note that all our results also hold for radius computation. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of ε-nets, region decomposition and other partition techniques.

When can we compute the diameter of a graph in quasi linear time? We address this question for the class of split graphs , that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either 2 or 3, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of a split graph in less than quadratic time. Therefore it is worth to study the complexity of diameter computation on subclasses of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded clique-interval number and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants and then almost completely settle the complexity of diameter computation on these subclasses of split graphs: For the k -clique-interval split graphs, we can compute their diameter in truly subquadratic time if \(k=\mathcal{O}(1)\) , and even in quasi linear time if \(k=o(\log {n})\) and in addition a corresponding ordering is given. However, under SETH this cannot be done in truly subquadratic time for any \(k = \omega (\log {n})\) . For the complements of k -clique-interval split graphs, we can compute their diameter in truly subquadratic time if \(k=\mathcal{O}(1)\) , and even in time \(\mathcal{O}(km)\) if a corresponding ordering is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a k -clique interval ordering can always be computed in quasi linear time. We prove that it is the case for \(k=1\) and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs – including the ones mentioned above – have a bounded clique-interval number. A research report version is deposited on HAL repository with number hal-02307397.

We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in L^2(G), the square of the line graph of G. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families with forbidden vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given a graph class \mathcal{G} characterized by a vertex ordering, and a graph G=(V,E) \in \mathcal{G} with a corresponding vertex ordering \sigma of V, one can produce (in linear time in the size of G) an ordering on the vertices of L^2(G), that shows that L^2(G) \in \mathcal{G} - for a number of graph classes \mathcal{G} - without computing the line graph or the square of the line graph of G. These results generalize and unify previous ones on showing closure under L^2(\cdot) for various graph families. Furthermore, these orderings on L^2(G) can be exploited algorithmically to compute a maximum induced matching on G faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, trapezoid graphs, and co-graphs, and we present the first \mathcal{O}(mn) time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known \mathcal{O}(n^4) time algorithm for the unweighted case.

In this paper we study how graph searching on a cocomparability graph $G$ can be used to produce cocomp orderings (i.e., orderings that are linear extensions of some transitive orientation of $\overline{G}$) that yield simple algorithms for various intractable problems in general. Such techniques have been used to find a simple certifying algorithm for the minimum path cover problem. In particular we present a characterization of the searches that preserve cocomp orderings when used as a “$^+$” sweep. This allows us to present a toolbox of different graph searches and a framework to solve various problems on cocomparability graphs. We illustrate these techniques by describing a very simple certifying algorithm for the maximum independent set problem as well as a simple permutation graph recognition algorithm.

For graph $G(V,E)$, a minimum path cover (MPC) is a minimum cardinality set of vertex disjoint paths that cover $V$ (i.e., every vertex of $G$ is in exactly one path in the cover). This problem is a natural generalization of the Hamiltonian path problem. Cocomparability graphs (the complements of graphs that have an acyclic transitive orientation of their edge sets) are a well studied subfamily of perfect graphs that includes many popular families of graphs such as interval, permutation, and cographs. Furthermore, for every cocomparability graph $G$ and acyclic transitive orientation of the edges of $\overline{G}$ there is a corresponding poset $P_G$; it is easy to see that an MPC of $G$ is a linear extension of $P_G$ that minimizes the bump number of $P_G$. Although there are directly graph-theoretical MPC algorithms (i.e., algorithms that do not rely on poset formulations) for various subfamilies of cocomparability graphs, notably interval graphs, until now all MPC algorithms for cocomparability graphs themselves have been based on the bump number algorithms for posets. In this paper we present the first directly graph-theoretical MPC algorithm for cocomparability graphs; this algorithm is based on two consecutive graph searches followed by a certifying algorithm. Surprisingly, except for a lexicographic depth first search (LDFS) preprocessing step, this algorithm is identical to the corresponding algorithm for interval graphs. The running time of the algorithm is $O({\rm min}(n^2, n + {\rm mloglogn}))$, with the nonlinearity coming from LDFS.

δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points u,v,w,x, the two larger of the sums d(u,v)+d(w,x), d(u,w)+d(v,x), d(u,x)+d(v,w) differ by at most 2δ. Given a finite set S of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of S with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for S with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δ-hyperbolic graphs and networks. Furthermore, we show that for δ-hyperbolic graphs G=(V,E) with uniformly bounded degrees of vertices, the exact center of S can be computed in linear time O(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δlog2 n). This construction has an additive error comparable with that given by Gromov for n-point δ-hyperbolic spaces, but can be implemented in O(|E|) time (instead of O(n2)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity.

Recently lexicographic breadth first search (LexBFS) has been shown to be a very powerful tool for the development of linear time, easily implementable recognition algorithms for various families of graphs. In this paper, we add to this work by producing a simple two LexBFS sweep algorithm to recognize the family of cographs. This algorithm extends to other related graph families such as $P_4$-reducible, $P_4$-sparse, and distance hereditary. It is an open question whether our cograph recognition algorithm can be extended to a similarly easy algorithm for modular decomposition.

Efficient implementation of type inclusion is an important feature of object oriented programming languages with multiple inheritance. The idea is to associate to each type a subset of a set S ={1,..., k } such that type inclusion coincides with subset inclusion. Such an embedding of types into 2 S (the lattice of all subsets of S ) is called a bit‐vector encoding of the type hierarchy. In this paper, we show that most known bit‐vector encoding methods can be inserted on a general theoretical framework using graph coloration, namely the notion of a simple encoding . We use the word simple because all these methods are heuristics for the general bit‐vector encoding problem, known as the 2‐dimension problem. First we provide a correct algorithm for partial orders based on simple encoding, improving the algorithm of Krall, Vitek, and Horspool (1997). Second we show that finding an optimal simple encoding is an NP‐hard problem. We end with a discussion on some practical issues.

Previous studies concerning multiple inheritance convinced us that a better analysis of conflict resolution mechanisms was necessary. In [DHHM92], we stated properties that a sound mechanism has to respect. Among them, a monotonicity principle plays a critical role, ensuring that the inheritance mechanism behaves “naturally” relative to the incremental design of the inheritance hierarchy. We focus here on linearizations and present an intrinsically monotonic linearization, whereas currently used linearizations are not. This paper describes the algorithm in detail, explains the design choices, and compares it to other linearizations, with LOOPS and CLOS taken as references. In particular, this new linearization extends CLOS and LOOPS linearizations, producing the same results when these linearizations are sound.

This paper investigates a transformation $P \to Q$ between partial orders $P,Q$ that transforms the interval dimension of P to the dimension of Q, i.e., $\text{idim} ( P ) = \dim ( Q )$. Such a construction has been shown before in the context of Ferrer’s dimension by Cogis [Discrete Math., 38 (1982), pp. 47–52]. The construction in this paper can be shown to be equivalent to his, but it has the advantage of (1) being purely order-theoretic, (2) providing a geometric interpretation of interval dimension similar to that of Ore [Amer. Math. Soc. Colloq. Publ., Vol. 38, 1962] for dimension, and (3) revealing several somewhat surprising connections to other order-theoretic results. For instance, the transformation $P \to Q$ can be seen as almost an inverse of the well-known split operation; it provides a theoretical background for the influence of edge subdivision on dimension (e.g., the results of Spinrad [Order, 5 (1989), pp. 143–147]) and interval dimension, and it turns out to be invariant with respect to changes of P that do not alter its comparability graph, thus also providing a simple new proof for the comparability invariance of interval dimension.

The main topic of this paper is multiple inheritance and conflict resolution methods in Object Oriented Programming. Our aim is to develop sound mechanisms easily understandable to any user. For this purpose, coherent behaviors of conflict resolution methods for multiple inheritance (such as supporting incrementality-monotonicity and stability under link subdivision) are introduced. We present interesting examples in which multiple inheritance known linearization algorithms (such as in CLOS [2] and LOOPS [19]) behave badly. Then we carefully study the conditions (on the inheritance graph) which assure good linearizations. We end with some suggestions for an incremental inheritance algorithm.