Nice labeling problem for event structures: a counterexample

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 $f: \mathbb{N} \mapsto \mathbb{N}$ such that an event structure with degree $\le n$ 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 Barth\'elemy and Constantin in 1993

1-Safe Petri nets and special cube complexes: equivalence and applications

Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net $N$ unfolds into an event structure $\mathcal{E}_N$. By a result of Thiagarajan (1996 and 2002), these unfoldings are exactly the trace regular event structures. Thiagarajan (1996 and 2002) conjectured that regular event structures correspond exactly to trace regular event structures. In a recent paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we proved that Thiagarajan's conjecture is true for regular event structures whose domains are principal filters of universal covers of (virtually) finite special cube complexes. In the current paper, 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 $\mathcal{E}_N$ (obtained as the unfolding of $N$) is a principal filter of the universal cover $\widetilde{X}_N$ of $X_N$. This establishes a bijection between 1-safe Petri nets and finite special cube complexes and provides a combinatorial characterization of trace regular event structures. Using this bijection and techniques from graph theory and geometry (MSO theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that the monadic second order logic of a 1-safe Petri net is decidable if and only if its unfolding is grid-free. Our counterexample is the trace regular event structure $\mathcal{\dot E}_Z$ which arises from a virtually special square complex $\dot Z$. The domain of $\mathcal{\dot E}_Z$ is grid-free (because it is hyperbolic), but the MSO theory of the event structure $\mathcal{\dot E}_Z$ is undecidable

A counterexample to Thiagarajan's conjecture on regular event structures

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 $\natural$-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. 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 $\natural$-cliques) admit a regular nice labeling. To disprove these conjectures, we describe a regular event domain (with bounded $\natural$-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 whose universal cover 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

On two conjectures of Maurer concerning basis graphs of matroids

We characterize 2-dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer (Conjecture 3 of S. Maurer, Matroid basis graphs I, JCTB 14 (1973), 216-240). We also establish Conjecture 1 from the same paper about the redundancy of the conditions in the characterization of basis graphs. We indicate positive-curvature-like aspects of the local properties of the studied complexes. We characterize similarly the corresponding 2-dimensional complexes of even $\Delta$-matroids.Comment: 28 page

Packing and covering with balls on Busemann surfaces

In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $\delta$, the minimum number of closed balls of radius $\delta$ with centers at $\mathcal S$ and covering the set $S$ is at most 19 times the maximum number of disjoint closed balls of radius $\delta$ centered at points of $S$: $\nu(S) \le \rho(S) \le 19\nu(S)$, where $\rho(S)$ and $\nu(S)$ are the covering and the packing numbers of $S$ by ${\delta}$-balls.Comment: 27 page

A multifacility location problem on median spaces

AbstractThis paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum-cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with ¦V¦points is O(¦V¦3 + ¦V¦ψ(n)), where ψ(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N + k(logN + logk + ψ(n))) time and O(N + kψ(n)) time in the case of the vertex-restricted multifacility location problem

Hypercellular graphs: partial cubes without Q3−Q_3^- as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.