302 research outputs found

### Reentrant transition in coupled noisy oscillators

We report on a novel type of instability observed in a noisy oscillator
unidirectionally coupled to a pacemaker. Using a phase oscillator model, we
find that, as the coupling strength is increased, the noisy oscillator lags
behind the pacemaker more frequently and the phase slip rate increases, which
may not be observed in averaged phase models such as the Kuramoto model.
Investigation of the corresponding Fokker-Planck equation enables us to obtain
the reentrant transition line between the synchronized state and the phase slip
state. We verify our theory using the Brusselator model, suggesting that this
reentrant transition can be found in a wide range of limit cycle oscillators.Comment: 16 pages, 7 figure

### Treedepth Parameterized by Vertex Cover Number

To solve hard graph problems from the parameterized perspective, structural parameters have commonly been used. In particular, vertex cover number is frequently used in this context. In this paper, we study the problem of computing the treedepth of a given graph G. We show that there are an O(tau(G)^3) vertex kernel and an O(4^{tau(G)}*tau(G)*n) time fixed-parameter algorithm for this problem, where tau(G) is the size of a minimum vertex cover of G and n is the number of vertices of G

### Parameterized Complexity of Graph Burning

Graph Burning asks, given a graph $G = (V,E)$ and an integer $k$, whether
there exists $(b_{0},\dots,b_{k-1}) \in V^{k}$ such that every vertex in $G$
has distance at most $i$ from some $b_{i}$. This problem is known to be
NP-complete even on connected caterpillars of maximum degree $3$. We study the
parameterized complexity of this problem and answer all questions arose by Kare
and Reddy [IWOCA 2019] about parameterized complexity of the problem. We show
that the problem is W[2]-complete parameterized by $k$ and that it does no
admit a polynomial kernel parameterized by vertex cover number unless
$\mathrm{NP} \subseteq \mathrm{coNP/poly}$. We also show that the problem is
fixed-parameter tractable parameterized by clique-width plus the maximum
diameter among all connected components. This implies the fixed-parameter
tractability parameterized by modular-width, by treedepth, and by distance to
cographs. Although the parameterization by distance to split graphs cannot be
handled with the clique-width argument, we show that this is also tractable by
a reduction to a generalized problem with a smaller solution size.Comment: 10 pages, 2 figures, IPEC 202

### Efficient Enumerations for Minimal Multicuts and Multiway Cuts

Let $G = (V, E)$ be an undirected graph and let $B \subseteq V \times V$ be a
set of terminal pairs. A node/edge multicut is a subset of vertices/edges of
$G$ whose removal destroys all the paths between every terminal pair in $B$.
The problem of computing a {\em minimum} node/edge multicut is NP-hard and
extensively studied from several viewpoints. In this paper, we study the
problem of enumerating all {\em minimal} node multicuts. We give an incremental
polynomial delay enumeration algorithm for minimal node multicuts, which
extends an enumeration algorithm due to Khachiyan et al. (Algorithmica, 2008)
for minimal edge multicuts. Important special cases of node/edge multicuts are
node/edge {\em multiway cuts}, where the set of terminal pairs contains every
pair of vertices in some subset $T \subseteq V$, that is, $B = T \times T$. We
improve the running time bound for this special case: We devise a polynomial
delay and exponential space enumeration algorithm for minimal node multiway
cuts and a polynomial delay and space enumeration algorithm for minimal edge
multiway cuts

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