We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hy... more We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube Q 3 (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VCdimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from being solved even for general set systems of VC-dimension 2. Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.
The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of... more The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of weighted distances from x to the vertices of G. For any integer p ≥ 2, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth power G p of G. This extends some characterizations of graphs with connected medians (case p = 1) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any p. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have G 2-connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.
The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatizati... more The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatization of the main classes of graphs occurring in Metric Graph Theory, in analogy to Tarski's axiomatization of Euclidean geometry. We provide such an axiomatization for weakly modular graphs and their principal subclasses (median and modular graphs, bridged graphs, Helly graphs, dual polar graphs, etc), basis graphs of matroids and even ∆-matroids, partial cubes and their subclasses (ample partial cubes, tope graphs of oriented matroids and complexes of oriented matroids, bipartite Pasch and Peano graphs, cellular and hypercellular partial cubes, almost-median graphs, netlike partial cubes), and Gromov hyperbolic graphs. On the other hand, we show that some classes of graphs (including chordal, planar, Eulerian, and dismantlable graphs), closely related with Metric Graph Theory, but defined in a combinatorial or topological way, do not allow such an axiomatization.
We examine connections between combinatorial notions that arise in machine learning and topologic... more We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that the previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous. On the positive side we present a new construction of an optimal unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabeled sample compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes.
The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the s... more The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the sum of distances from x to all vertices of P. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the 1-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure, due to their bijections with CAT(0) cube complexes and domains of event structures. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges (Θ-classes or hyperplanes) via Lexicographic Breadth First Search (LexBFS). To prove the correctness of our algorithm, we show that any LexBFS ordering of the vertices of G satisfies the following fellow traveler property of independent interest: the parents of any two adjacent vertices of G are also adjacent. Using the fast computation of the Θ-classes, we also compute the Wiener index (total distance) of G in linear time and the distance matrix in optimal quadratic time.
Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty inte... more Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs -- Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graphical C(4)$-$T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type $C_n$ are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right-angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and...
The median of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances f... more The median of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all other vertices of $G$. It is known that computing the median of dense graphs in subcubic time refutes the APSP conjecture and computing the median of sparse graphs in subquadratic time refutes the HS conjecture. In this paper, we present a linear time algorithm for computing medians of median graphs, improving over the existing quadratic time algorithm. Median graphs constitute the principal class of graphs investigated in metric graph theory, due to their bijections with other discrete and geometric structures (CAT(0) cube complexes, domains of event structures, and solution sets of 2-SAT formulas). Our algorithm is based on the known majority rule characterization of medians in a median graph $G$ and on a fast computation of parallelism classes of edges ($\Theta$-classes) of $G$. The main technical contribution of the paper is a linear time algorithm for computing the $\...
Memoirs of the American Mathematical Society, 2020
United States. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more ... more United States. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number. Back number information. For back issues see www.ams.org/backvols.
Distance labeling schemes are schemes that label the vertices of a graph with short labels in suc... more Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such a way that given the labels of a source node and a destination node, it is possible to compute efficiently the port number of the edge from the source that heads in the direction of the destination. One of important problems is finding natural classes of graphs admitting distance and/or routing labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on n nodes enjoys distance and routing labeling schemes with labels of O(log 3 n) bits.
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular event str... more We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular event structures correspond exactly to event structures obtained as unfoldings of finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999) that domains of regular event structures with bounded-cliques are recognizable by finite trace automata. Event structures, trace automata, and Petri nets are fundamental models in concurrency theory. There exist nice interpretations of these structures as combinatorial and geometric objects and both conjectures can be reformulated in this framework. Namely, from a graph theoretical point of view, the domains of prime event structures correspond exactly to median graphs; from a geometric point of view, these domains are in bijection with CAT(0) cube complexes. A necessary condition for both conjectures to be true is that domains of regular event structures (with bounded-cliques) admit a regular nice labeling (which corresponds to a special coloring of the hyperplanes of the associated CAT(0) cube complex). To disprove these conjectures, we describe a regular event domain (with bounded-cliques) that does not admit a regular nice labeling. Our counterexample is derived from an example by Wise (1996 and 2007) of a nonpositively curved square complex X with six squares, whose edges are colored in five colors, and whose universal cover X is a CAT(0) square complex containing a particular plane with an aperiodic tiling. We prove that other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009). On the positive side, using breakthrough results by Agol (2013) and Haglund and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's conjecture is true for regular event structures whose domains occur as principal filters of hyperbolic CAT(0) cube complexes which are universal covers of finite nonpositively curved cube complexes.
We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hy... more We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from...
Nielsen et al. [35] proved that every 1-safe Petri net N unfolds into an event structure E N . By... more Nielsen et al. [35] proved that every 1-safe Petri net N unfolds into an event structure E N . By a result of Thiagarajan [46], these unfoldings are exactly the trace-regular event structures. Thiagarajan [46] conjectured that regular event structures correspond exactly to trace-regular event structures. In a recent paper (Chalopin and Chepoi [12]), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. However, we proved that Thiagarajan’s conjecture is true for regular event structures whose domains are principal filters of universal covers of finite special cube complexes. In the current article, we prove the converse: To any finite 1-safe Petri net N , one can associate a finite special cube complex X N such that the domain of the event structure E N (obtained as the unfolding of N ) is a principal filter of the universal cover X̃ N of X N . This establishes a bijection between 1-safe Petri net...
We give a characterization of distance-preserving subgraphs of Johnson graphs, i.e. of graphs whi... more We give a characterization of distance-preserving subgraphs of Johnson graphs, i.e. of graphs which are isometrically embeddable into Johnson graphs (the Johnson graph J(m, Λ) has the subsets of cardinality m of a set Λ as the vertex-set and two such sets A, B are adjacent iff |A B| = 2). Our characterization is similar to the characterization of D.Ž. Djoković (J. Combin. Th. Ser. B 14 (1973), 263-267) of distancepreserving subgraphs of hypercubes and provides an explicit description of the wallspace (split system) defining the embedding.
Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarde... more Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic l1-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group D4), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.
Surveys on Discrete and Computational Geometry, 2008
The article surveys structural characterizations of several graph classes defined by distance pro... more The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fiber-complemented graphs, or l 1-graphs. Several kinds of l 1-graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆-)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or tree-like graphs such as distance-hereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the graph classes reported in this survey.
In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move... more In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move at different speeds s and s with s < s, are δ-hyperbolic with δ = O(s 2). We also show that the dependency between δ and s is linear if s − s = Ω(s) and G obeys a slightly stronger condition. This solves an open question from the paper J. Chalopin et al., Cop and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25 (2011) 333-359. Since any δ-hyperbolic graph is cop-win for s = 2r and s = r + 2δ for any r > 0, this establishes a new-game-theoretical-characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between δ and s is linear for any s < s. Using these results, we describe a simple constant-factor approximation of the hyperbolicity δ of a graph on n vertices in O(n 2) time when the graph is given by its distance-matrix.
In this article, we design optimal or near optimal interval routing schemes (IRS, for short) with... more In this article, we design optimal or near optimal interval routing schemes (IRS, for short) with small compactness for several classes of plane quadrangulations and triangulations (by optimality or near optimality we mean that messages are routed via shortest or almost shortest paths). We show that the subgraphs of the rectilinear grid bounded by simple circuits allow optimal IRS with at most two circular intervals per edge (2-IRS). We extend this result to all plane quadrangulations in which all inner vertices have degrees ¿ 4. Namely, we establish that every such graph has an optimal IRS with at most seven linear intervals per edge (7-LIRS). This leads to a 7-LIRS with the stretch factor 2 for all plane triangulations in which all inner vertices have degrees ¿ 6. All routing schemes can be implemented in linear time.
Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrila... more Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard. Finite squaregraphs can be recognized in linear time by a Breadth-First-Search.
In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (cal... more In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function f : N → N such that an event structure with degree ≤ n admits a labeling with at most f (n) labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barthélemy and Constantin in 1993.
We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hy... more We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube Q 3 (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VCdimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from being solved even for general set systems of VC-dimension 2. Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.
The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of... more The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of weighted distances from x to the vertices of G. For any integer p ≥ 2, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth power G p of G. This extends some characterizations of graphs with connected medians (case p = 1) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any p. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have G 2-connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.
The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatizati... more The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatization of the main classes of graphs occurring in Metric Graph Theory, in analogy to Tarski's axiomatization of Euclidean geometry. We provide such an axiomatization for weakly modular graphs and their principal subclasses (median and modular graphs, bridged graphs, Helly graphs, dual polar graphs, etc), basis graphs of matroids and even ∆-matroids, partial cubes and their subclasses (ample partial cubes, tope graphs of oriented matroids and complexes of oriented matroids, bipartite Pasch and Peano graphs, cellular and hypercellular partial cubes, almost-median graphs, netlike partial cubes), and Gromov hyperbolic graphs. On the other hand, we show that some classes of graphs (including chordal, planar, Eulerian, and dismantlable graphs), closely related with Metric Graph Theory, but defined in a combinatorial or topological way, do not allow such an axiomatization.
We examine connections between combinatorial notions that arise in machine learning and topologic... more We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that the previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous. On the positive side we present a new construction of an optimal unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabeled sample compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes.
The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the s... more The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the sum of distances from x to all vertices of P. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the 1-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure, due to their bijections with CAT(0) cube complexes and domains of event structures. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges (Θ-classes or hyperplanes) via Lexicographic Breadth First Search (LexBFS). To prove the correctness of our algorithm, we show that any LexBFS ordering of the vertices of G satisfies the following fellow traveler property of independent interest: the parents of any two adjacent vertices of G are also adjacent. Using the fast computation of the Θ-classes, we also compute the Wiener index (total distance) of G in linear time and the distance matrix in optimal quadratic time.
Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty inte... more Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs -- Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graphical C(4)$-$T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type $C_n$ are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right-angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and...
The median of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances f... more The median of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all other vertices of $G$. It is known that computing the median of dense graphs in subcubic time refutes the APSP conjecture and computing the median of sparse graphs in subquadratic time refutes the HS conjecture. In this paper, we present a linear time algorithm for computing medians of median graphs, improving over the existing quadratic time algorithm. Median graphs constitute the principal class of graphs investigated in metric graph theory, due to their bijections with other discrete and geometric structures (CAT(0) cube complexes, domains of event structures, and solution sets of 2-SAT formulas). Our algorithm is based on the known majority rule characterization of medians in a median graph $G$ and on a fast computation of parallelism classes of edges ($\Theta$-classes) of $G$. The main technical contribution of the paper is a linear time algorithm for computing the $\...
Memoirs of the American Mathematical Society, 2020
United States. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more ... more United States. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number. Back number information. For back issues see www.ams.org/backvols.
Distance labeling schemes are schemes that label the vertices of a graph with short labels in suc... more Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such a way that given the labels of a source node and a destination node, it is possible to compute efficiently the port number of the edge from the source that heads in the direction of the destination. One of important problems is finding natural classes of graphs admitting distance and/or routing labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on n nodes enjoys distance and routing labeling schemes with labels of O(log 3 n) bits.
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular event str... more We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular event structures correspond exactly to event structures obtained as unfoldings of finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999) that domains of regular event structures with bounded-cliques are recognizable by finite trace automata. Event structures, trace automata, and Petri nets are fundamental models in concurrency theory. There exist nice interpretations of these structures as combinatorial and geometric objects and both conjectures can be reformulated in this framework. Namely, from a graph theoretical point of view, the domains of prime event structures correspond exactly to median graphs; from a geometric point of view, these domains are in bijection with CAT(0) cube complexes. A necessary condition for both conjectures to be true is that domains of regular event structures (with bounded-cliques) admit a regular nice labeling (which corresponds to a special coloring of the hyperplanes of the associated CAT(0) cube complex). To disprove these conjectures, we describe a regular event domain (with bounded-cliques) that does not admit a regular nice labeling. Our counterexample is derived from an example by Wise (1996 and 2007) of a nonpositively curved square complex X with six squares, whose edges are colored in five colors, and whose universal cover X is a CAT(0) square complex containing a particular plane with an aperiodic tiling. We prove that other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009). On the positive side, using breakthrough results by Agol (2013) and Haglund and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's conjecture is true for regular event structures whose domains occur as principal filters of hyperbolic CAT(0) cube complexes which are universal covers of finite nonpositively curved cube complexes.
We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hy... more We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from...
Nielsen et al. [35] proved that every 1-safe Petri net N unfolds into an event structure E N . By... more Nielsen et al. [35] proved that every 1-safe Petri net N unfolds into an event structure E N . By a result of Thiagarajan [46], these unfoldings are exactly the trace-regular event structures. Thiagarajan [46] conjectured that regular event structures correspond exactly to trace-regular event structures. In a recent paper (Chalopin and Chepoi [12]), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. However, we proved that Thiagarajan’s conjecture is true for regular event structures whose domains are principal filters of universal covers of finite special cube complexes. In the current article, we prove the converse: To any finite 1-safe Petri net N , one can associate a finite special cube complex X N such that the domain of the event structure E N (obtained as the unfolding of N ) is a principal filter of the universal cover X̃ N of X N . This establishes a bijection between 1-safe Petri net...
We give a characterization of distance-preserving subgraphs of Johnson graphs, i.e. of graphs whi... more We give a characterization of distance-preserving subgraphs of Johnson graphs, i.e. of graphs which are isometrically embeddable into Johnson graphs (the Johnson graph J(m, Λ) has the subsets of cardinality m of a set Λ as the vertex-set and two such sets A, B are adjacent iff |A B| = 2). Our characterization is similar to the characterization of D.Ž. Djoković (J. Combin. Th. Ser. B 14 (1973), 263-267) of distancepreserving subgraphs of hypercubes and provides an explicit description of the wallspace (split system) defining the embedding.
Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarde... more Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic l1-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group D4), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.
Surveys on Discrete and Computational Geometry, 2008
The article surveys structural characterizations of several graph classes defined by distance pro... more The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fiber-complemented graphs, or l 1-graphs. Several kinds of l 1-graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆-)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or tree-like graphs such as distance-hereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the graph classes reported in this survey.
In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move... more In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move at different speeds s and s with s < s, are δ-hyperbolic with δ = O(s 2). We also show that the dependency between δ and s is linear if s − s = Ω(s) and G obeys a slightly stronger condition. This solves an open question from the paper J. Chalopin et al., Cop and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25 (2011) 333-359. Since any δ-hyperbolic graph is cop-win for s = 2r and s = r + 2δ for any r > 0, this establishes a new-game-theoretical-characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between δ and s is linear for any s < s. Using these results, we describe a simple constant-factor approximation of the hyperbolicity δ of a graph on n vertices in O(n 2) time when the graph is given by its distance-matrix.
In this article, we design optimal or near optimal interval routing schemes (IRS, for short) with... more In this article, we design optimal or near optimal interval routing schemes (IRS, for short) with small compactness for several classes of plane quadrangulations and triangulations (by optimality or near optimality we mean that messages are routed via shortest or almost shortest paths). We show that the subgraphs of the rectilinear grid bounded by simple circuits allow optimal IRS with at most two circular intervals per edge (2-IRS). We extend this result to all plane quadrangulations in which all inner vertices have degrees ¿ 4. Namely, we establish that every such graph has an optimal IRS with at most seven linear intervals per edge (7-LIRS). This leads to a 7-LIRS with the stretch factor 2 for all plane triangulations in which all inner vertices have degrees ¿ 6. All routing schemes can be implemented in linear time.
Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrila... more Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard. Finite squaregraphs can be recognized in linear time by a Breadth-First-Search.
In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (cal... more In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function f : N → N such that an event structure with degree ≤ n admits a labeling with at most f (n) labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barthélemy and Constantin in 1993.
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Papers by Victor Chepoi