6,102 research outputs found
Zero forcing number, constrained matchings and strong structural controllability
The zero forcing number is a graph invariant introduced to study the minimum
rank of the graph. In 2008, Aazami proved the NP-hardness of computing the zero
forcing number of a simple undirected graph. We complete this NP-hardness
result by showing that the non-equivalent problem of computing the zero forcing
number of a directed graph allowing loops is also NP-hard. The rest of the
paper is devoted to the strong controllability of a networked system. This kind
of controllability takes into account only the structure of the interconnection
graph, but not the interconnection strengths along the edges. We provide a
necessary and sufficient condition in terms of zero forcing sets for the strong
controllability of a system whose underlying graph is a directed graph allowing
loops. Moreover, we explain how our result differs from a recent related result
discovered by Monshizadeh et al. Finally, we show how to solve the problem of
finding efficiently a minimum-size input set for the strong controllability of
a self-damped system with a tree-structure.Comment: Submitted as a journal paper in May 201
Zero Forcing Sets and Bipartite Circulants
In this paper we introduce a class of regular bipartite graphs whose
biadjacency matrices are circulant matrices and we describe some of their
properties. Notably, we compute upper and lower bounds for the zero forcing
number for such a graph based only on the parameters that describe its
biadjacency matrix. The main results of the paper characterize the bipartite
circulant graphs that achieve equality in the lower bound.Comment: 22 pages, 13 figure
Zero forcing in iterated line digraphs
Zero forcing is a propagation process on a graph, or digraph, defined in
linear algebra to provide a bound for the minimum rank problem. Independently,
zero forcing was introduced in physics, computer science and network science,
areas where line digraphs are frequently used as models. Zero forcing is also
related to power domination, a propagation process that models the monitoring
of electrical power networks.
In this paper we study zero forcing in iterated line digraphs and provide a
relationship between zero forcing and power domination in line digraphs. In
particular, for regular iterated line digraphs we determine the minimum
rank/maximum nullity, zero forcing number and power domination number, and
provide constructions to attain them. We conclude that regular iterated line
digraphs present optimal minimum rank/maximum nullity, zero forcing number and
power domination number, and apply our results to determine those parameters on
some families of digraphs often used in applications
A Unifying Framework for Strong Structural Controllability
This paper deals with strong structural controllability of linear systems. In
contrast to existing work, the structured systems studied in this paper have a
so-called zero/nonzero/arbitrary structure, which means that some of the
entries are equal to zero, some of the entries are arbitrary but nonzero, and
the remaining entries are arbitrary (zero or nonzero). We formalize this in
terms of pattern matrices whose entries are either fixed zero, arbitrary
nonzero, or arbitrary. We establish necessary and sufficient algebraic
conditions for strong structural controllability in terms of full rank tests of
certain pattern matrices. We also give a necessary and sufficient graph
theoretic condition for the full rank property of a given pattern matrix. This
graph theoretic condition makes use of a new color change rule that is
introduced in this paper. Based on these two results, we then establish a
necessary and sufficient graph theoretic condition for strong structural
controllability. Moreover, we relate our results to those that exists in the
literature, and explain how our results generalize previous work.Comment: 11 pages, 6 Figure
Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC
We have observations concerning the set theoretic strength of the following
combinatorial statements without the axiom of choice. 1. If in a partially
ordered set, all chains are finite and all antichains are countable, then the
set is countable. 2. If in a partially ordered set, all chains are finite and
all antichains have size , then the set has size
for any regular . 3. CS (Every partially
ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF
(Every partially ordered set has a cofinal well-founded subset). 5. DT
(Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If
the chromatic number of a graph is finite (say ), and the
chromatic number of another graph is infinite, then the chromatic
number of is . 7. For an infinite graph and a finite graph , if every finite subgraph of
has a homomorphism into , then so has . Further we study a few statements
restricted to linearly-ordered structures without the axiom of choice.Comment: Revised versio
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
Dynamic approach to k-forcing
The k-forcing number of a graph is a generalization of the zero forcing
number. In this note, we give a greedy algorithm to approximate the k-forcing
number of a graph. Using this dynamic approach, we give corollaries which
improve upon two theorems from a recent paper of Amos, Caro, Davila and Pepper
[2], while also answering an open problem posed by Meyer [9]
- …