2,757 research outputs found
Graph Colorings with Constraints
A graph is a collection of vertices and edges, often represented by points and connecting lines in the plane. A proper coloring of the graph assigns colors to the vertices, edges, or both so that proximal elements are assigned distinct colors. Here we examine results from three different coloring problems. First, adjacent vertex distinguishing total colorings are proper total colorings such that the set of colors appearing at each vertex is distinct for every pair of adjacent vertices. Next, vertex coloring total weightings are an assignment of weights to the vertices and edges of a graph so that every pair of adjacent vertices have distinct weight sums. Finally, edge list multi-colorings consider assignments of color lists and demands to edges; edges are colored with a subset of their color list of size equal to its color demand so that adjacent edges have disjoint sets. Here, color sets consisting of measurable sets are considered
Strong Structural Controllability of Systems on Colored Graphs
This paper deals with structural controllability of leader-follower networks.
The system matrix defining the network dynamics is a pattern matrix in which a
priori given entries are equal to zero, while the remaining entries take
nonzero values. The network is called strongly structurally controllable if for
all choices of real values for the nonzero entries in the pattern matrix, the
system is controllable in the classical sense. In this paper we introduce a
more general notion of strong structural controllability which deals with the
situation that given nonzero entries in the system's pattern matrix are
constrained to take identical nonzero values. The constraint of identical
nonzero entries can be caused by symmetry considerations or physical
constraints on the network. The aim of this paper is to establish graph
theoretic conditions for this more general property of strong structural
controllability.Comment: 13 page
On the Complexity of Distributed Splitting Problems
One of the fundamental open problems in the area of distributed graph
algorithms is the question of whether randomization is needed for efficient
symmetry breaking. While there are fast, -time randomized
distributed algorithms for all of the classic symmetry breaking problems, for
many of them, the best deterministic algorithms are almost exponentially
slower. The following basic local splitting problem, which is known as the
\emph{weak splitting} problem takes a central role in this context: Each node
of a graph has to be colored red or blue such that each node of
sufficiently large degree has at least one node of each color among its
neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly
simple problem is complete w.r.t. the above fundamental open question in the
following sense: If there is an efficient -time determinstic
distributed algorithm for weak splitting, then there is such an algorithm for
all locally checkable graph problems for which an efficient randomized
algorithm exists. In this paper, we investigate the distributed complexity of
weak splitting and some closely related problems. E.g., we obtain efficient
algorithms for special cases of weak splitting, where the graph is nearly
regular. In particular, we show that if and are the minimum
and maximum degrees of and if , weak splitting can
be solved deterministically in time
. Further, if and , there is a
randomized algorithm with time complexity
The Weisfeiler-Leman Dimension of Planar Graphs is at most 3
We prove that the Weisfeiler-Leman (WL) dimension of the class of all finite
planar graphs is at most 3. In particular, every finite planar graph is
definable in first-order logic with counting using at most 4 variables. The
previously best known upper bounds for the dimension and number of variables
were 14 and 15, respectively.
First we show that, for dimension 3 and higher, the WL-algorithm correctly
tests isomorphism of graphs in a minor-closed class whenever it determines the
orbits of the automorphism group of any arc-colored 3-connected graph belonging
to this class.
Then we prove that, apart from several exceptional graphs (which have
WL-dimension at most 2), the individualization of two correctly chosen vertices
of a colored 3-connected planar graph followed by the 1-dimensional
WL-algorithm produces the discrete vertex partition. This implies that the
3-dimensional WL-algorithm determines the orbits of a colored 3-connected
planar graph.
As a byproduct of the proof, we get a classification of the 3-connected
planar graphs with fixing number 3.Comment: 34 pages, 3 figures, extended version of LICS 2017 pape
Trade-Offs in Distributed Interactive Proofs
The study of interactive proofs in the context of distributed network computing is a novel topic, recently introduced by Kol, Oshman, and Saxena [PODC 2018]. In the spirit of sequential interactive proofs theory, we study the power of distributed interactive proofs. This is achieved via a series of results establishing trade-offs between various parameters impacting the power of interactive proofs, including the number of interactions, the certificate size, the communication complexity, and the form of randomness used. Our results also connect distributed interactive proofs with the established field of distributed verification. In general, our results contribute to providing structure to the landscape of distributed interactive proofs
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