13,620 research outputs found
1-Safe Petri nets and special cube complexes: equivalence and applications
Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net
unfolds into an event structure . 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
one can associate a finite special cube complex such that the
domain of the event structure (obtained as the unfolding of
) is a principal filter of the universal cover of .
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
which arises from a virtually special square complex . The domain of
is grid-free (because it is hyperbolic), but the MSO
theory of the event structure is undecidable
Radio numbers for generalized prism graphs
A radio labeling is an assignment such that
every distinct pair of vertices satisfies the inequality
d(u,v)+|c(u)-c(v)|\geq \diam(G)+1. The span of a radio labeling is the
maximum value. The radio number of , , is the minimum span over all
radio labelings of . Generalized prism graphs, denoted , , , have vertex set and
edge set .
In this paper we determine the radio number of for and .
In the process we develop techniques that are likely to be of use in
determining radio numbers of other families of graphs.Comment: To appear in Discussiones Mathematicae Graph Theory. 16 pages, 1
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Classifying Families of Character Degree Graphs of Solvable Groups
We investigate prime character degree graphs of solvable groups. In
particular, we consider a family of graphs constructed by
adjoining edges between two complete graphs in a one-to-one fashion. In this
paper we determine completely which graphs occur as the prime
character degree graph of a solvable group.Comment: 7 pages, 5 figures, updated version is reorganize
Special Graphs
A special p-form is a p-form which, in some orthonormal basis {e_\mu}, has
components \phi_{\mu_1...\mu_p} = \phi(e_{\mu_1},..., e_{\mu_p}) taking values
in {-1,0,1}. We discuss graphs which characterise such forms.Comment: 8 pages, V2: a texing error correcte
Note on group distance magic graphs
A \emph{group distance magic labeling} or a \gr-distance magic labeling of
a graph with is an injection from to an Abelian
group \gr of order such that the weight of
every vertex is equal to the same element \mu \in \gr, called the
magic constant. In this paper we will show that if is a graph of order
for some natural numbers , such that \deg(v)\equiv c
\imod {2^{p+1}} for some constant for any , then there exists
an \gr-distance magic labeling for any Abelian group \gr for the graph
. Moreover we prove that if \gr is an arbitrary Abelian group of
order such that \gr \cong \zet_2 \times\zet_2 \times \gA for some
Abelian group \gA of order , then exists a \gr-distance magic labeling
for any graph
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